diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzogub" "b/data_all_eng_slimpj/shuffled/split2/finalzzogub" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzogub" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\n\n\\subsection{}\n By a \\emph{Fourier quasicrystal} one often means a \ndiscrete measure, whose Fourier transform is also a \ndiscrete measure.\n This concept was inspired by the experimental\n discovery in the middle of 80's of non-periodic \n atomic structures with diffraction patterns consisting\n of spots. In this context, different versions of ``discreteness'' were discussed,\n see in particular \\cite{bom}, \\cite{cah}, \\cite{mey2}, \\cite{dys}.\n\\par\n Let $\\mu$ be a (complex) measure on $\\R^n$, supported on a discrete set $\\Lambda$: \n \\begin{equation}\n\t \\mu =\\sum_{\\lambda\\in \\Lambda} \\mu(\\lambda)\\delta_{\\lambda}, \\quad\n\t \\mu(\\lambda)\\ne 0.\n\t \\label{RM.1}\n \\end{equation}\nWe shall suppose that $\\mu$ is a slowly increasing measure, which means that\n$|\\mu|\\{x:|x|1$.\n\\par\nMoreover, in our construction both $\\mu$ and $\\ft\\mu$ \nare translation-bounded measures (see Section \\ref{sec:prelim} for the\n definition). This implies that \n$\\mu$ is an almost periodic measure, whose Fourier\ntransform $\\ft\\mu$ is also an almost periodic measure.\nBy definition, a measure $\\mu$ on $\\R^n$ is an \\define{almost periodic\nmeasure} if for every continuous, compactly supported function\n$\\varphi$ on $\\R^n$, the convolution $\\mu \\ast \\varphi$ is an\nalmost periodic function in the sense of Bohr.\n\n\n\n\\subsection{}\nIn the crystallography community, it seems to be commonly agreed\n that the support $\\Lambda$ should be a uniformly discrete set.\n We remind that Meyer's quasicrystals \\cite{mey2},\n which appeared first under the name ``model sets'' \\cite{mey0},\n are uniformly discrete sets, and they support measures\n whose spectra are dense countable sets.\n\\par\n So it is a natural problem, to what extent can the spectrum $S$ of a non-periodic \nquasi\\-crystal be discrete, assuming that the support $\\Lambda$ is uniformly discrete?\n In the present paper we address this problem, and consider quasicrystals with non-symmetric\ndiscreteness assumptions on the support and the spectrum.\n\\par\nFirst we obtain several results which show that, under certain conditions,\nif the spectrum $S$ is a discrete closed set, then in fact $S$ must be uniformly discrete.\nThese results thus reduce the situation to the setting in \\thmref{RT.0} above, which in turn\nallows us to conclude that the measure $\\mu$ is representable in the form\n\\eqref{RM.3}.\n\\par\nOn the other hand, we present an example of a non-periodic quasicrystal such\nthat the spectrum $S$ is a nowhere dense countable set.\n\\par\nWe also apply our results to Hof's quasicrystals. In this context, we prove that\nif a Delone set $\\Lam$ of finite local complexity has a uniformly discrete diffraction spectrum,\nthen the diffraction measure of $\\Lam$ has a periodic structure.\n\\par\nFinally, we extend our results to the more general situation, where $\\hat{\\mu}$\n is a measure which has both a pure point component and a continuous one.\n\n\n\n\n\n\\section{Results}\n\n\\subsection{}\n Our first result deals with the case when $\\mu$ is a positive-definite measure: \n \\begin{thm}\n\t \\label{RT.5}\n\t Let $\\mu$ be a positive-definite measure on $\\R^n$\n\t satisfying \\eqref{RM.1} and \\eqref{RM.2}. Assume that the support $\\Lambda$ is a\n\t uniformly discrete set, while the spectrum $S$ is a discrete closed set. Then\n\t also $S$ is uniformly discrete, and the measure $\\mu$ has the form \\eqref{RM.3}. \n \\end{thm}\n\t Actually, we will show that if a positive-definite\n\t measure $\\mu$ with uniformly discrete support $\\Lambda$ is not of the form\n\t \\eqref{RM.3}, then its spectrum $S$ must have a relatively dense set of\n\t accumulation points (see \\thmref{AT.5}). \n\n\\subsection{}\nIn the next result, which holds for dimension $n=1$, the\n\t positive-definiteness assumption is replaced by a stronger discreteness condition\n\t on the spectrum:\n\\begin{thm}\n\t\\label{RT.6}\n\tLet $\\mu$ be a measure on $\\R$ satisfying \\eqref{RM.1} and \\eqref{RM.2}. Assume\n\tthat the support $\\Lambda$ is a uniformly discrete set, while\n the spectrum $S$ satisfies the condition\n\t \\begin{equation}\n \\sup_{x\\in\\R}\\# (S \\cap [x,x+1]) < \\infty. \n\t\t \\label{RM.4}\n\t \\end{equation}\nThen\t$S$ is a uniformly discrete set, and $\\mu$ is of the form \\eqref{RM.3}. \n\\end{thm}\nNotice that condition \\eqref{RM.4} means that $S$ is the union of a finite number of uniformly discrete sets. \n\n\\subsection{}\nIn the following result, a stronger discreteness condition is imposed on the support:\n\\begin{thm}\n\t\\label{RT.7}\n\tLet $\\mu$ be a measure on $\\R^n$ satisfying \\eqref{RM.1} and\n\t\\eqref{RM.2}.\n\tAssume that the set $\\Lambda-\\Lambda$ is uniformly discrete, and that\n\t$S$ is a discrete closed set. Then the same conclusion as in the previous two theorems holds. \n\\end{thm}\n\t Moreover, we will prove that if $\\Lambda-\\Lambda$ is uniformly discrete, but the\n\t measure $\\mu$ is not of the form\n\t \\eqref{RM.3}, then again the spectrum $S$ must have a relatively dense set of\n\t accumulation points (see \\thmref{CT.1}).\n\\par\n\\thmref{RT.7} implies the following result communicated to us by Y.\\ Meyer:\na ``simple quasicrystal'' cannot support a measure whose spectrum\nis a discrete closed set (for the definition of a simple quasicrystal, see \\cite{matei-meyer-simple}).\n\n \n\\subsection{}\n The previous results show that if the measure $\\mu$\ndoes not have the periodic structure \\eqref{RM.3},\n then the spectrum $S$ must have finite accumulation points.\n However, $S$ need not be dense in any ball, as the following result shows:\n\\begin{thm}\n\t\\label{RT.15}\n\tThere is a positive-definite measure $\\mu$ on $\\R^n$\nsatisfying \\eqref{RM.1} and \\eqref{RM.2}, such that:\n\\begin{enumerate-math}\n\\item\nThe support $\\Lam$ is not contained in a finite union of translates of any lattice;\n\\item\nThe set $\\Lam-\\Lam$ is uniformly discrete; \n\\item\nThe spectrum $S$ is a nowhere dense (countable) set.\n\\end{enumerate-math}\n\\end{thm}\nTo prove this we base on Meyer's construction,\nbut with an additional modification.\n\n\n\n\\subsection{}\nOur results can be applied to quasicrystals in Hof's sense \\cite{hof}.\n\\par\nRecall that a set $\\Lambda\\subset\\R^n$ is called a \\emph{Delone set} if it is uniformly discrete and also\nrelatively dense. A Delone set $\\Lambda$ is said to be of \\emph{finite local complexity} if the difference\nset $\\Lambda-\\Lambda$ is a discrete closed set. This means that $\\Lambda$ has, up to translations,\nonly a finite number of local patterns of any given size. \n\\par\nAn \\emph{autocorrelation measure} $\\gamma_\\Lam$ of the set $\\Lambda$ is any weak limit point of the\nmeasures \n\t \\begin{equation}\n (2R)^{-n}\\sum_{\\lambda,\\lambda'\\in \\Lam \\cap [-R,R]^{n}}\\delta_{\\lambda'-\\lambda}\n\t\t \\label{RT.8.1}\n\t \\end{equation}\nas $R\\to\\infty$. An autocorrelation measure $\\gamma_\\Lam$ is always\npositive-definite, and so its Fourier transform $\\hat{\\gamma}_\\Lam$ is a positive\nmeasure, called a \\emph{diffraction measure} of $\\Lambda$. If the measure\n$\\hat{\\gamma}_\\Lambda$ is purely atomic, then \nits support $S$ is called a\n\\define{diffraction spectrum} of $\\Lambda$.\n\\par\nThe following result answers a question posed in \\cite[Problem 4.2(a)]{lag2}:\n\\begin{thm}\n\t\\label{RT.8}\nSuppose that\n\\begin{enumerate-math}\n\\item\n $\\Lambda\\subset\\R^n$ is a Delone set of finite local complexity;\n\\item\n\t$\\gamma_\\Lam$ is an autocorrelation measure of $\\Lambda$; and\n\\item\n\tThe diffraction spectrum $S$ (the support of $\\hat{\\gamma}_\\Lam$) is a uniformly discrete set. \n\\end{enumerate-math}\nThen $S$ is contained in a finite union of translates of some lattice,\nand the diffraction measure $\\hat{\\gamma}_\\Lambda$ has the form \\eqref{RM.3}.\n\\end{thm}\nA slightly more general version of this result will be given in \\thmref{DT.1}.\nThe same conclusion is true if the set $\\Lambda-\\Lambda$ is uniformly discrete, \nand the diffraction spectrum $S$ is a discrete closed set (see \\thmref{DT.2}).\n\n\n\\subsection{}\nWe also consider discrete measures $\\mu$, whose\nFourier transform $\\hat{\\mu}$ is a measure which has both a pure point\ncomponent and a continuous one. We can extend our previous results\nto this more general situation using the following:\n\\begin{thm}\n\t\\label{RT.10} \n\tLet $\\mu$ be a measure on $\\R^n$ with uniformly discrete\nsupport $\\Lambda$, and assume that $\\hat{\\mu}$ is a (slowly increasing)\nmeasure. Then the discrete part of $\\hat{\\mu}$ is the Fourier transform\nof another measure $\\mu'$, whose support $\\Lambda'$ is also a uniformly discrete set.\n\\end{thm}\nMoreover, if $\\Lambda-\\Lambda$ is a uniformly discrete set,\nthen also $\\Lambda'-\\Lambda'$ is uniformly discrete.\n\\par\nBy applying the previous results to this new measure $\\mu'$, one can obtain\nversions of the results for measures $\\mu$ with non pure point Fourier transform\n(see \\thmref{FT.2}).\n\n \n\n\n\\section{Preliminaries}\n\\label{sec:prelim}\n\n\\subsection{Notation}\nBy $\\dotprod{\\cdot}{\\cdot}$ and $|\\cdot|$ we denote the Euclidean \nscalar product and norm in $\\R^n$. \nThe open ball of radius $r$ centered at the origin is denoted\n $B_r := \\{x \\in \\R^n: |x| 0.\n\\end{equation}\nThe set $\\Lambda$ is said to be \\emph{relatively dense} if there is $R > 0$ such that every ball of\nradius $R$ intersects $\\Lam$. \n\\par\nBy a (full-rank) lattice $L \\subset \\R^n$ we mean the image of $\\Z^n$ under some\ninvertible linear transformation $T$. The determinant $\\det(L)$ is equal to $|\\det (T)|$.\nThe dual lattice $L^*$ is the set of all vectors $\\lambda^*$ such that $\\dotprod{\\lambda}{\\lambda^*}\n \\in \\Z$, $\\lambda \\in L$.\n\\par\nBy a ``distribution'' we will mean a temperate distribution on $\\R^n$.\nBy a ``measure'' we mean a complex, locally finite measure (usually infinite) which is\nassumed to be \\emph{slowly increasing}. By definition, a measure $\\mu$ is slowly\nincreasing if there is a constant $N$ such that \n$|\\mu|(B_R)=O(R^N)$ as $R\\to \\infty.$\nThe measure $\\mu$ is called \\emph{translation-bounded} if\n\\begin{equation}\\label{tr-bdd-def}\n\\sup_{x \\in \\R^n} |\\mu|(x+B_1) < \\infty.\n\\end{equation}\n\\par\nAny translation-bounded measure is slowly increasing, and any\nslowly increasing measure is a temperate distribution. Remark\nthat for a \\emph{positive} measure to be a temperate distribution, it is \nalso necessary to be slowly increasing, but this is not true\nfor complex (or real, signed) measures.\n\\par\nBy the ``support'' of a pure point measure $\\mu$ we mean the \ncountable set of the non-zero atoms of $\\mu$. This should not be confused with the\nnotion of support in the sense of distributions, which is always a closed set.\n\\par\nThe Fourier transform on $\\R^n$ will be normalized as follows:\n$$\\ft \\varphi (t)=\\int_{\\R^n} \\varphi (x) \\, e^{-2\\pi i\\langle t,x\\rangle} dx.$$\n\\par\nWe denote by $\\supp(\\varphi)$ the closed support of a Schwartz function $\\varphi$,\nand by $\\spec (\\varphi)$ the closed support of its Fourier transform $\\ft\\varphi$.\n\\par\nIf $\\alpha$ is a temperate distribution, and $\\varphi$ is a Schwartz function on $\\R^n$,\nthen $\\dotprod{\\alpha}{\\varphi}$ will denote the action of $\\alpha$ on ${\\varphi}$.\nThe Fourier transform $\\ft\\alpha$ of the distribution $\\alpha$ is defined by \n$\\dotprod{\\ft\\alpha}{\\varphi} = \\dotprod{\\alpha}{\\ft\\varphi}$.\n\\par\nA distribution $\\alpha$ is called \\emph{positive} if $\\dotprod{\\alpha}{\\varphi} \\geq 0$\nfor any Schwartz function $\\varphi \\geq 0$. It is well-known that if $\\alpha$ is a positive distribution,\nthen it is a positive measure. A distribution $\\alpha$ is called \\emph{positive-definite} if $\\ft\\alpha$ is a positive \ndistribution. \n\\par\nFor a set $A \\subset \\R^n$ we denote by $\\# A$ the number of elements in $A$, and\nby $\\mes(A)$ or $|A|$ the Lebesgue measure of $A$.\n\n\n\\subsection{Measures}\nWe will need some basic facts about measures in $\\R^n$.\n\\begin{lem}\n\t\\label{AL.4}\n\tLet $\\mu$ be a measure on $\\R^n$, whose support $\\Lambda$ is a uniformly\n\tdiscrete set. Assume that $\\ft{\\mu}$ is a slowly increasing measure. Then \n\t\\begin{equation}\n\t \\sup_{\\lambda\\in\\Lambda} |\\mu(\\lambda)|<\\infty,\n\t\t\\label{AL.4.1}\n\t\\end{equation}\n\tand so $\\mu$ is a translation-bounded measure. \n\\end{lem}\nThis can be proved in a similar way to \\cite[Lemma 2]{lo2}.\n\\begin{lem}\n\t\\label{AL.10}\nLet $\\mu$ be a measure on $\\R^n$, whose support $\\Lambda$ is a uniformly discrete\nset. Assume that $\\hat{\\mu}$ is a slowly increasing measure, with at least one non-zero\natom. Then $\\Lambda$ is a relatively dense set in $\\R^n$. \n\\end{lem}\n\\begin{proof}\n\tBy Lemma \\ref{AL.4}, the measure $\\mu$ is translation-bounded. Let us suppose that\n\t$\\Lambda$ is not relatively dense, and show that this implies that $\\hat{\\mu}(\\{s\\})=0$\nfor every $s \\in \\R^n$.\n\\par\n\tChoose a Schwartz function $\\varphi$ whose Fourier transform $\\hat{\\varphi}$ has compact\n\tsupport, and $\\hat{\\varphi}(0)=1$. For each $0<\\delta<1$ define\n\t\t$\\varphi_\\delta(t):= \\delta^n\\varphi(\\delta t)$. Then we have \n\t\\begin{equation}\n\t\t\\hat{\\mu}(\\{s\\})=\\lim_{\\delta\\to 0}\\int \\hat{\\varphi}_\\delta(t-s)e^{2\\pi\n\t\ti\\dotprod{x}{t-s} }d\\hat\\mu(t)\n\t\t\\label{AL.10.1}\n\t\\end{equation}\n\tuniformly with respect to $x\\in\\R^n$. On the other hand, \n\t\\begin{equation}\n\t\t\\int \\hat{\\varphi}_\\delta(t-s)e^{2\\pi i\\dotprod{x}{t-s}}d\\hat{\\mu}(t) = \n\t\t\\int \\varphi_\\delta(x-y)e^{-2\\pi i\\dotprod{s}{y}}d\\mu(y).\n\t\t\\label{AL.10.2}\n\t\\end{equation}\n\tIf $\\Lambda$ is not relatively dense, then for any $R>0$ there is $x\\in \\R^n$ such\n\tthat the ball $x+B_{R}$ does not intersect $\\Lambda$. Using the\n\ttranslation-boundedness of $\\mu$ this implies that for any $\\delta>0$, there are\n\tvalues of $x$ for which the right-hand side of \\eqref{AL.10.2} is arbitrarily\n\tclose to zero. Hence the limit in \\eqref{AL.10.1} must be zero, which proves the\n\tclaim.\n\\end{proof}\n\n\n\n\\subsection{Interpolation}\nFor a compact set $\\Omega \\subset \\R^n$, we denote by $\\B(\\Omega)$ the Bernstein space\nconsisting of all bounded, continuous functions $f$ on $\\R^n$ such that the distribution $\\ft f$\nis supported by $\\Omega$. \nA set $\\Lam\\subset\\R^n$ is called an \\emph{interpolation set} for the space $\\B(\\Omega)$ if for every bounded sequence\n $\\{c_\\lam\\}$, $\\lam\\in\\Lam$, there exists at least one $f \\in \\B(\\Omega)$ satisfying\n$f(\\lam)=c_\\lam$ $(\\lam\\in\\Lam)$. It is well-known that such\n$\\Lam$ must be a uniformly discrete set.\n\\par\nThe following result is due to Ingham for $n=1$, and Kahane for $n>1$, see\n\\cite{ou3}. \n\\begin{thm}\n\tThere is a constant $C$ which depends on the dimension $n$ only, such that if\n\t$\\Lambda$ is a uniformly discrete set in $\\R^n$, $d(\\Lambda)\\ge a> 0$, then\n\t$\\Lambda$ is an interpolation set for $\\mathcal{B}(\\Omega)$ where $\\Omega$ is any closed\n\tball of radius $C\/a$. \n\t\\label{AL.11}\n\\end{thm}\nAs a consequence we obtain:\n\\begin{corollary}\n\t\\label{AC.12}\n\tThere is a constant $C$ which depends on the dimension $n$ only, such that if a measure\n\t$\\mu$ is supported on a uniformly discrete set $\\Lambda\\subset \\R^n$,\n\t$d(\\Lambda)\\ge a> 0$, and if the distribution $\\hat{\\mu}$ vanishes on a ball of radius $C\/a$, then $\\mu=0$. \n\\end{corollary}\n\n\n\\begin{proof}\nSuppose that $\\hat{\\mu}$ vanishes on the ball $B_R$, where $R:=(C+1)\/a$ and\n$C$ is the constant from \\thmref{AL.11}\n (we may assume that the ball is centered at the origin). \nLet $\\Omega=\\{x : |x| \\leq C\/a\\}$. \nGiven $\\lambda\\in\\Lambda$, there is $f\\in \\mathcal{B}(\\Omega)$ such that $f(\\lambda)=1$ and $f$ vanishes on\n\t$\\Lambda\\setminus \\{\\lambda\\}$. Define $\\varphi(x):= f(x)\\psi(x)$, where $\\psi$ is a\n\tSchwartz function such that $\\psi(\\lambda)=1$ and $\\spec(\\psi)\\subset B_{1\/a}$.\n\tThen $\\varphi$ is a Schwartz function satisfying\n\\[\n\\varphi(\\lambda)=1, \\quad\n\t\\text{$\\varphi(\\lambda')=0$ for all $\\lambda' \\in \\Lam \\setminus \\{\\lam\\}$}, \\quad\n\\spec(\\varphi)\\subset B_R.\n\\]\nHence\n\\[ \\mu(\\lambda)=\\int\n\t\\overline{\\varphi(x)}d\\mu(x)=\\dotprod{\\hat{\\mu}}{\\overline{\\hat{\\varphi}}}=0.\n\\]\n\tThis holds for any $\\lambda\\in\\Lambda$, so we obtain $\\mu=0$. \n\\end{proof}\n\n\n\n\n\\section{Auxiliary measures $\\nu_h$}\nLet $\\mu$ be a measure on $\\R^n$ satisfying \\eqref{RM.1} and \\eqref{RM.2},\nand assume that the support $\\Lambda$ is a uniformly discrete set.\nBy Lemma \\ref{AL.4}, the atoms of $\\mu$ are bounded, so $\\mu$ is a translation-bounded measure. \n\n For each $h\\in S-S$ we denote \n\\begin{equation}\n\tS_h:= S\\cap (S-h)= \\left\\{ s\\in S\\,:\\, s+h\\in S \\right\\}, \n\t\\label{AL.3.1}\n\\end{equation}\nwhich is a non-empty subset of $S$, and we introduce a new measure \n\\begin{equation}\n\t\\nu_{h}:= \\sum_{s\\in S_h}\\hat{\\mu}(s)\\,\\overline{\\hat{\\mu}(s+h)}\\,\\delta_s.\n\t\\label{AL.3.2}\n\\end{equation}\nNotice that it is a non-zero, slowly increasing measure, whose support is the set $S_h$.\n\\begin{lem}\n\t\\label{AL.6}\n\tThe Fourier transform $\\hat{\\nu}_h$ of the measure $\\nu_{h}$ is also a measure,\n\twhich is translation-bounded and supported by the closure of the set\n\t$\\Lambda-\\Lambda$.\n\\end{lem}\n\nThis is an elaborated version of \\cite[Lemma 12]{lo2}. That lemma stated that $\\spec(\\nu_h)$\ndoes not intersect the punctured ball $B_a\\setminus\\left\\{ 0 \\right\\}$, where $a:=d(\\Lambda)>0$.\nHowever, the result there was formulated with the roles of\n\t$\\Lambda$ and $S$ interchanged, and under the stronger assumption that $\\Lambda$,\n\t$S$ are both uniformly discrete sets. \n\n\t\\begin{proof}[Proof of Lemma \\ref{AL.6}] \n\t\tFix a Schwartz function $\\varphi$, whose Fourier transform $\\hat{\\varphi}$ has\n\t\tcompact support and $\\hat{\\varphi}(0)=1$. For each $0<\\delta<1$ we denote\n\t\t$\\varphi_\\delta(x):= \\delta^n\\varphi(\\delta x)$, and define the measure\n\t\t\\begin{equation}\n\t\t\t\\nu_h^{(\\delta)} (t) := \\overline{\\left( \\hat{\\mu} \\ast \\hat{\\varphi}_\\delta\n\t\t\t\\right)(t+h)}\\cdot\n\t\t\t\\hat{\\mu}(t).\n\t\t\t\\label{AL.6.1}\n\t\t\\end{equation}\nIt is a slowly increasing measure, supported by $S$, which tends to $\\nu_h$ in the space of temperate\ndistributions as $\\delta\\to 0$. Hence $\\hat{\\nu}_h^{(\\delta)}$ tends to $\\hat{\\nu}_h$ in the\nsame sense as $\\delta\\to 0$. \n\\par\nWe will show that $\\hat{\\nu}_h^{(\\delta)}$ is a\ntranslation-bounded measure, and moreover \n\\[ \\sup_{x\\in\\R^n} |\\hat{\\nu}_{h}^{(\\delta)}| (x+B_1) \\] \nis bounded by some constant $C(\\mu,\\varphi)$ which depends on $\\mu$ and $\\varphi$ only\n(in particular, it does not depend on $\\delta$).\n Indeed, the Fourier transform of $\\nu_h^{(\\delta)}$ is the\nmeasure \n\\begin{equation}\n\t\\hat{\\nu}_h^{(\\delta)}= \\overline{(\\varphi_\\delta\\cdot e_{-h}\\cdot \\mu)(x)} \\ast\n\t\\mu(-x),\n\t\\label{AL.6.2}\n\\end{equation}\nwhere $e_{-h}(x):= e^{-2\\pi i \\dotprod{h}{x}}$. Hence, we have\n\\[ \\sup_{x\\in \\R^n} |\\hat{\\nu}_h^{(\\delta)}| (x+B_1) \\le \n\\Big\\{ \\sup_{x\\in \\R^n} |\\mu| (x+B_1) \\Big\\} \\Big\\{\n\\int |\\varphi_\\delta (x)| \\, |d\\mu(x)| \\Big\\} \\le C(\\mu,\\varphi), \\]\nsince $\\mu$ is translation-bounded. Letting $\\delta\\to 0$ this implies\nthat the limit $\\hat{\\nu}_h$ is also a translation-bounded measure,\nand in fact $|\\hat{\\nu}_h| (x+B_1) \\le C(\\mu,\\varphi)$ for all $x\\in \\R^n$.\n\\par\nFinally, it follows from \\eqref{AL.6.2} that the measure $\\hat{\\nu}_h^{(\\delta)}$ is supported by\n $\\Lambda-\\Lambda$. Hence its limit as $\\delta\\to 0$ must be supported by the closure\n of $\\Lambda-\\Lambda$, which ends the proof. \n\t\\end{proof}\n\n\n\n\n\\section{Positive-definite measures}\n\\label{sec:pos-def}\n\\subsection{ } \\label{5.1a}\nIn this section we consider positive-definite measures whose supports are\nuniformly discrete sets. We establish a dichotomy concerning the\ndiscreteness of the spectrum: either it is also uniformly discrete, or it is\n``non-discrete'' in a strong sense. \n\\begin{thm}\n\t\\label{AT.5}\n\tLet $\\mu$ be a positive-definite measure on $\\R^n$ satisfying \\eqref{RM.1} and\n\t\\eqref{RM.2}, and assume that the support $\\Lambda$ is a uniformly discrete set. Then, either \n\\begin{enumerate-math}\n\\item\n$S$ is also uniformly discrete; or \n\\item\n $S$ has a relatively dense set of accumulation points.\n\\end{enumerate-math}\n\\end{thm}\n\nIn particular, it follows that if the spectrum $S$ is a discrete closed set, then it must\nbe uniformly discrete. Hence \\thmref{RT.0} applies to this situation, and yields that the\nmeasure $\\mu$ is representable in the form \\eqref{RM.3}. So \\thmref{RT.5} follows.\n\n\\begin{remark}\nActually we will prove that there is a constant\n\t$C$ which depends on the dimension $n$ only, such that if the spectrum\n$S$ is not uniformly discrete, then every ball of radius\n\t$C\/d(\\Lam)$ contains infinitely many points of $S$.\n\\end{remark}\n\n\n\\subsection{}\nWe will need the following auxiliary lemma:\n\\begin{lem}\n\t\\label{AL.1}\n\tThere is a real-valued Schwartz function $\\varphi$ on $\\R^n$ which has the following\n\tproperties: \n\t\\begin{enumerate-math}\n\t\\item \\label{it_al.1.1} There is $R$ such that $\\varphi(x)>0$ for $|x|\\ge R$;\n\t\\item \\label{it_al.1.2} $\\int \\varphi(x)dx=0$;\n\t\\item \\label{it_al.1.3} $ \\spec(\\varphi)$ is contained in $B_1$ (the open unit ball).\n\t\\end{enumerate-math}\n\\end{lem}\n\\begin{proof}\n\tChoose a Schwartz function $\\psi > 0$ whose Fourier transform $\\hat{\\psi}$ is\n\tsupported in the ball $\\left\\{t: |t|\\le 1\/3 \\right\\}$. Define $\\varphi:=\\alpha \\psi\n\t-\\beta \\psi^2$, where $\\alpha:= \\left( \\int \\psi \\right)^{-1}$ and $\\beta:=\\left(\n\t\\int \\psi^2 \\right)^{-1}$. It is easy to verify that all the properties\n\t\\ref{it_al.1.1}, \\ref{it_al.1.2} and \\ref{it_al.1.3} are satisfied.\n\\end{proof} \n\\subsection{}\n\\begin{proof}[Proof of \\thmref{AT.5}]\n\tAssume that $S$ is not uniformly discrete. Let $\\varphi$ be the function given by\n\t\\lemref{AL.1}, and $R$ be the number from property \\ref{it_al.1.1} of that lemma.\n\tWe will show that \n\tany closed ball of radius $C\/a$ contains infinitely many points of $S$, where\n$C : = R+1$ and $a:=d(\\Lambda)>0$ (notice that the constant $C$ indeed depends on the dimension $n$ only). \nIn particular this will show that the set of accumulation points of $S$ must be relatively dense.\n\n\\par\n\tBy multiplying $\\mu$ by an exponential (which corresponds to translation of\n\t$\\hat{\\mu}$), it will be enough to show that the ball $\\{|t| \\leq C\/a\\}$\ncontains infinitely many points of $S$. So, suppose to the\n\tcontrary that this does not hold, namely this ball\n\tcontains only finitely many points of $S$. \n\\par\n Since $S$ is not uniformly discrete,\n\tthe set $S-S$ contains elements $h\\neq 0$ arbitrarily close to zero. Hence we may choose\n\t$h\\in S-S$ such that the set $S_{h}$ defined by \\eqref{AL.3.1} does not intersect\n\tthe ball $\\left\\{ |t| < R\/a \\right\\}$. \tIt follows that the measure $\\nu_h$\n\tin \\eqref{AL.3.2} is a non-zero positive measure, whose support is\n\tcontained in $\\left\\{ |t|\\ge R\/a \\right\\}$. Using property \\ref{it_al.1.1} from\n\t\\lemref{AL.1} this implies that \n\t\\begin{equation}\n\t\t\\label{AT.5.1}\n\t\t\\int \\varphi(ax)\\,d\\nu_h(x) > 0.\n\t\\end{equation}\n\tOn the other hand, we have \n\t\\begin{equation}\n\t\t\\label{AT.5.2}\n\t\t\\int\n\t\t\\varphi(ax)\\,d\\nu_h(x)=a^{-n}\\int \\hat{\\varphi}(-t\/a) \\, d\\hat{\\nu}_h(t).\n\t\\end{equation}\n\tBy \\lemref{AL.6}, $\\hat{\\nu}_h$ is a measure, supported by the closure of\n$\\Lam-\\Lam$. Since $\\Lam$ is uniformly discrete, \n this closure is contained in the set $\\{0\\}\\cup\n\t\\{|t|\\ge a\\}$.\n But from properties \\ref{it_al.1.2} and \\ref{it_al.1.3} in\n\t\\lemref{AL.1} it follows that the function $\\hat{\\varphi}(-t\/a)$ vanishes on this set.\n\tHence, the right-hand side of \\eqref{AT.5.2} must vanish, \n\tin contradiction with \\eqref{AT.5.1}. \n\\end{proof}\n\n\n\n\\section{Spectra with finite density}\nIn this section we prove \\thmref{RT.6}. We will show that under the conditions in the\ntheorem, the spectrum $S$ of the measure $\\mu$ must be a uniformly discrete set. Then the final conclusion that\n$\\mu$ is of the form \\eqref{RM.3} can be deduced from \\thmref{RT.0}. \n\\subsection{}\nFor a set $\\Lambda\\subset \\R $ we denote \n\\[ \\rho(\\Lambda):= \\sup_{x\\in\\R}\\# (\\Lambda\\cap [x,x+1]). \\] \nNotice that $\\rho(\\Lambda)<\\infty$ if and only if $\\Lambda$ is a finite union of uniformly\ndiscrete sets. \n\\par\nWe will need the following notion of ``lower density'' of a set $\\Lambda \\subset \\R$, defined by \n\\[ D_{\\#}(\\Lambda):=\\liminf_{R\\to\\infty}\\frac{\\#(\\Lambda\\cap (-R,R))}{2R}. \\]\nClearly we have $D_{\\#}(\\Lambda)\\le\\rho(\\Lambda)$. It will be useful below to extend the\ndefinition of the density $D_{\\#}$ also to multi-sets $\\Lambda \\subset \\R$, that is, to the\ncase when points in $\\Lambda$ occur with multiplicities. Notice that $D_{\\#}$ is\nsuper-additive in the sense that \n\\[ D_{\\#}(A\\cup B)\\ge D_{\\#}(A)+D_{\\#}(B), \\]\nwhere the union $A\\cup B$ is understood in the sense of multi-sets.\n\\subsection{ }\nThe following result is a more general version of \\cite[Proposition 4]{lo2}.\n\\begin{prop}\n\t\\label{BP.1}\n\tLet $\\Lambda\\subset \\R$ be a set with $\\rho(\\Lambda)\\le M< \\infty$. Assume that\n\t$\\Lambda$ supports a non-zero, slowly increasing measure $\\mu$, such that the distribution\n\t$\\hat{\\mu}$ vanishes on the open interval $(0,a)$ for some $a> 0$. Then \n\t\\[ D_{\\#}(\\Lambda)\\ge c(a,M), \\]\n\twhere $c(a,M)>0$ is a constant which depends on $a$ and $M$ only. \n\\end{prop}\nThis can be deduced from the following lemma:\n\\begin{lem}\n\t\\label{BL.2}\n\tLet $\\Lambda$ be a finite subset of $(-R,R)\\setminus (-1,1)$, such that\n\t$\\rho(\\Lambda)\\le M$, and let $a>0$. There is $c(a,M)>0$ such that if\n\t$(\\#\\Lambda)\/(2R)0$ such that $D_{\\#}(S_h)>c$ for every $h\\in S-S$. Then\n\t$\\rho(S-S)<\\infty$. \n\\end{lem}\n\\begin{proof}\n\tLet $x\\in\\R$, and suppose that $h_1,\\dots, h_N$ are distinct points in the set\n\t$(S-S)\\cap [x,x+1]$. Since the lower density $D_{\\#}$ is super-additive, we have \n\t\\[ cN \\le \\sum_{j=1}^{N}D_{\\#}(S_{h_j})\\le D_{\\#}\\Big(\n\t\\bigcup_{j=1}^{N}S_{h_j} \\Big), \\]\nwhere the union is understood in the sense of multi-sets. Notice that each point in this\nunion occurs with multiplicity not greater than $\\rho(S)$. It follows that $cN\\le\n\\rho(S) D_{\\#}(S)$, which shows that the set $S-S$ cannot have more than\n$\\rho(S) D_{\\#}(S)\/c$ elements in any closed interval of length 1. Hence $\\rho(S-S)<\\infty$,\nwhich proves the claim.\n\\end{proof}\n\\subsection{}\n\\begin{proof}[Proof of \\thmref{RT.6}] \nWe assume that $\\mu$ is a measure on $\\R$ satisfying \\eqref{RM.1} and\n\\eqref{RM.2}, where $\\Lambda$ is a uniformly discrete set, and $S$ is a set with\n$\\rho(S)<\\infty$.\n\\par\n For each $h\\in S-S$, let $\\nu_{h}$ be the measure\ndefined by \\eqref{AL.3.2}. By \\lemref{AL.6} the Fourier transform $\\hat{\\nu}_h$ is\nsupported by the closure of the set $\\Lambda-\\Lambda$. Since $\\Lambda$ is uniformly\ndiscrete this implies that $\\hat{\\nu}_{h}$ vanishes on the open interval $(0,a)$, where\n$a:=d(\\Lambda)>0.$ As the measure $\\nu_h$ is supported by $S_h$, it follows from \\propref{BP.1} that \n$ D_{\\#}(S_h)\\ge c$, where $c>0$ is a constant which depends on $d(\\Lambda)$ and $\\rho(S)$. \n\\par\nSince this holds for every $h\\in S-S$, \\lemref{BL.3} allows us to deduce that $\\rho(S-S)<\\infty$. In particular, the set\n$S-S$ has no accumulation point at zero, so there is $\\delta>0$ such that $(S-S)\\cap\n(-\\delta,\\delta)=\\left\\{ 0 \\right\\}$. Hence $S$ must be uniformly discrete, and\nin fact $d(S)\\ge \\delta$. \n\\par\nOnce we have concluded that \n$\\Lambda$ and $S$ are both uniformly discrete sets, we can apply \\thmref{RT.0} which yields that the\nmeasure $\\mu$ is representable in the form \\eqref{RM.3}.\n\\end{proof}\n\n\n\n\\section{Meyer sets}\n\\subsection{}\nIn this section we show that if the support $\\Lam$ satisfies a stronger \n discreteness condition than in \\thmref{AT.5}, then the conclusion of this theorem\n remains true without any additional positivity restriction.\n\\begin{definition*}\n\tA set $\\Lambda\\subset \\R^n$ is called a \\emph{Delone set} if $\\Lambda$ is both a\n\tuniformly discrete and relatively dense set. \n\\end{definition*}\n\\lemref{AL.10} implies that a uniformly discrete set $\\Lam$ which supports a measure $\\mu$,\nwhose Fourier transform $\\ft\\mu$ is a pure point\nmeasure, must be a Delone set.\n\\begin{definition*}\n\tA set $\\Lambda\\subset \\R^n$ is called a \\emph{Meyer set} if the following two conditions\n\tare satisfied: \n\t\\begin{enumerate-math}\n\t\\item $\\Lambda$ is a Delone set;\n\t\\item There is a finite set $F$ such that $\\Lambda-\\Lambda\\subset \\Lambda+F$. \n\t\\end{enumerate-math}\n\\end{definition*}\nThe concept of Meyer set was introduced in \\cite{mey0, mey1} in connection with problems\nin harmonic analysis. After the experimental discovery of quasicrystalline materials in\nthe middle of 80's, Meyer sets have been extensively studied as mathematical models of\nquasicrystals. \n\\par\nThere are some equivalent forms of the definition of a Meyer set,\nsee \\cite{moo}. In particular, the following is true (Lagarias \\cite{lag1}):\n\\par\n\\emph{A Delone set $\\Lam$ is a Meyer set if and only if\n$\\Lambda-\\Lambda$ is uniformly discrete}\n\\par\n\\noindent\n (a simplified version of the proof of this equivalence can be found in \\cite[Lemma 8]{lo2}).\n \n\n\\subsection{}\nNow we show that if a measure $\\mu$ is supported by a Meyer set $\\Lam$, then the\ndichotomy phenomenon for the spectrum $S$ is valid: either $S$ is uniformly discrete, or\nit is non-discrete with a relatively dense set of accumulation points. \n\\begin{thm}\n\t\\label{CT.1}\n\tLet $\\mu$ be a measure on $\\R^n$ satisfying \\eqref{RM.1} and \\eqref{RM.2},\n\tand assume that the support $\\Lam$ is a Meyer set. Then the same\n\tconclusion as in \\thmref{AT.5} holds.\n\\end{thm}\n\n\n\\begin{proof}\nAs in the proof of \\thmref{AT.5}, it will\n\tbe enough to prove that for every $h\\in S-S$, the set $S_h$ must intersect any ball\n\tof radius $C\/a$, but now we will take $C$ to be the number from \\corref{AC.12}, and\n\t$a:=d(\\Lambda-\\Lambda)>0$.\n\\par\nAnd indeed, by \\lemref{AL.6} the measure $\\nu_h$\n\tdefined by \\eqref{AL.3.2} is a non-zero measure, whose Fourier transform\n\t$\\hat{\\nu}_h$ is a translation-bounded measure supported by the set\n\t$\\Lambda-\\Lambda$ (this set is uniformly discrete, so it is not necessary to consider\nits closure). Now \\corref{AC.12}, applied to the measure\n\t$\\hat{\\nu}_h$, implies that $\\nu_h$ cannot vanish on a ball of radius\n\t$C\/a$. Hence $S_h$ must intersect any such a ball, which completes the proof.\n\\end{proof}\n\n\\begin{remark}\nThe proof in fact shows that there is a constant\n\t$C$ which depends on the dimension $n$ only, such that if the spectrum\n$S$ is not uniformly discrete, then every ball of radius\n\t$C\/d(\\Lam-\\Lam)$ contains infinitely many points of $S$.\n\\end{remark}\n\n\n\\subsection{}\nNext, we deduce \\thmref{RT.7} from the above result. Suppose that $\\Lambda-\\Lambda$ is a\nuniformly discrete set, and that $S$ is a discrete closed set. Hence $S$ \nhas no finite accumulation points, so it follows from \\thmref{CT.1} that $S$\n must be uniformly discrete. \n\\par\nHowever, to complete the conclusion of \\thmref{RT.7} it still remains to show that $\\mu$ \nis representable in the form \\eqref{RM.3}. Here one cannot directly\napply \\thmref{RT.0}, since the measure was assumed to be neither positive nor positive-definite. \nIn order to obtain \\eqref{RM.3} we use instead the following version of\n\\thmref{RT.0}, proved in \\cite{lo1}:\n\\begin{thm}\n\t\\label{RT.9}\n\tLet $\\mu$ be a measure on $\\R^n$ satisfying \\eqref{RM.1} and \\eqref{RM.2}. \n\tAssume that the sets $\\Lambda-\\Lambda$ and $S$ are both uniformly discrete. Then the\n\tconclusion of \\thmref{RT.0} holds.\n\\end{thm}\n Combining Theorems \\ref{CT.1} and \\ref{RT.9} thus implies the full assertion of \\thmref{RT.7}.\n\n\n\n\n\n\n\n\\section{Nowhere dense spectra}\n\n\\label{subsec:notdense}\nIn this section we prove \\thmref{RT.15}. We will construct a non-periodic\nmeasure $\\mu$ supported on a Meyer set $\\Lam\\subset \\R^n$,\nsuch that the spectrum $S$ is not dense in any ball. \nMoreover, the measure $\\mu$ in the construction \n is positive-definite, and both $\\mu$ and $\\ft\\mu$\nare translation-bounded measures.\n\n\n\n\\subsection{}\n Let $\\Gamma$ be a lattice in $\\R^n\\times \\R^m$, and let\n\t$p_1$ and $p_2$ denote the projections onto $\\R^n$ and $\\R^m$ respectively. Assume\n\tthat the restrictions of $p_1$ and $p_2$ to $\\Gamma$ are injective, and that their\n\timages are dense in $\\R^n$ and $\\R^m$ respectively.\n\tLet $\\Gamma^{*}$ be the dual\n\tlattice, then the restrictions of $p_1$ and $p_2$ to $\\Gamma^{*}$ are also\n\tinjective and have dense images.\n\\par\n If $\\Omega$ is a bounded open set in\n\t$\\R^m$, then the set \n\t\\begin{equation}\n\\Lambda(\\Gamma,\\Omega):= \\left\\{ p_1(\\gamma)\\,:\\,\n\t\\gamma\\in \\Gamma, \\, p_2(\\gamma)\\in \\Omega \\right\\} \n\t\t\\label{RP.9.0}\n\t\\end{equation}\nis called the ``model set'',\n\tor the ``cut-and-project set'', associated to the lattice $\\Gamma$ and to the\n\t``window'' $\\Omega$. \nIt is well-known that any model set is a Meyer set.\n\\par\nMeyer observed \\cite[p.\\ 30]{mey0} (see also \\cite{mey2}) that\n\tmodel sets provide examples of non-periodic uniformly discrete sets, which support\n\ta measure $\\mu$ such that the Fourier transform $\\ft\\mu$ is also a pure point measure. Such a\n\tmeasure may be obtained by choosing a Schwartz function $\\varphi$ on\n$\\R^m$ such that $\\supp(\\hat{\\varphi})\\subset \\Omega$, and taking \n\t\\begin{equation}\n\t\t\\mu = \\sum_{\\gamma\\in\\Gamma} \\hat{\\varphi} ( p_2(\\gamma) )\n\t\t\\delta_{p_1(\\gamma)}.\n\t\t\\label{RP.9.1}\n\t\\end{equation}\n\tIt is not difficult to verify that this is a translation-bounded measure, whose Fourier transform is the\n\t(also translation-bounded) pure point measure\n\t\\begin{equation}\n\t\t\\hat{\\mu}=\\frac{1}{\\det \\Gamma}\n\t\t\\sum_{\\gamma^{*}\\in\\Gamma^{*}}\\varphi( p_2(\\gamma^{*})\n\t\t)\\delta_{p_1(\\gamma^{*})}.\n\t\t\\label{RP.9.2}\n\t\\end{equation}\n\\par\nHowever, the compact support of $\\hat{\\varphi}$ implies that $\\varphi$ is an entire function,\nand so $\\varphi$ cannot also be supported on a bounded set. Hence\n the spectrum of the measure $\\mu$ is only known to be contained in $p_1(\\Gam^*)$, \nand so it is generally everywhere dense in $\\R^n$.\n\\par\nNevertheless, we will see that one can construct a function $\\varphi$ with sufficiently many zeros,\nin such a way that the non-zero atoms in \\eqref{RP.9.2} in fact lie in a nowhere dense set.\n\n\\subsection{} \\label{section_8.2}\nA set $\\Lambda\\subset\\R^n$ is said to have a \\emph{uniform density} $D(\\Lambda)$ if \n\\[ \\frac{ \\#\\left( \\Lambda \\cap (x+B_R) \\right)}{|B_R|} \\to D(\\Lambda) \\]\nas $R\\to \\infty$ uniformly with respect to $x\\in \\R^n$. \n\\begin{lem}\n\t\\label{EL.2}\n\tLet $\\Lambda=\\Lambda(\\Gamma,\\Omega)$ be a model set such that the boundary of\n\t$\\Omega$ is a set of Lebesgue measure zero in $\\R^m$. Then $\\Lambda$ has uniform\n\tdensity \n\t\\[ D(\\Lambda) = \\frac{\\mes(\\Omega)}{\\det(\\Gamma)}. \\]\n\\end{lem}\nA proof of this fact can be found e.g.\\ in \\cite[Proposition 5.1]{matei-meyer-simple}. \n\\subsection{} \\label{8 .3}\nWe will now assume that $m=1$, that is, $\\Gam$ is a lattice in $\\R^n \\times \\R$.\nThe following theorem is the main result of this section.\n\\begin{thm}\n\t\\label{ET.1}\n\tFor any $\\varepsilon > 0$ there is a non-zero Schwartz function $\\varphi\\ge 0$ on\n\t$\\R$, such that: \n\t\\begin{enumerate-math}\n\t\\item The measure $\\mu$ in \\eqref{RP.9.1} is supported by the model set\n\t\t$\\Lambda(\\Gamma,(-\\varepsilon,\\varepsilon))$;\n\t\\item The spectrum of $\\mu$ is a nowhere dense set in $\\R^n$. \n\t\\end{enumerate-math}\n\\end{thm}\nObserve that the support of $\\mu$ cannot be covered by a finite \nunion of translates of any lattice, since it contains a model set. Hence this \nresult implies \\thmref{RT.15}. The condition $\\varphi\\ge 0$ guarantees that $\\mu$ is a\npositive-definite measure. \n\\par\nThe proof of \\thmref{ET.1} depends on the following: \n\\begin{lem}\n\t\\label{EL.3}\n\tFor each $j\\ge 1$ let $Q_j\\subset\\R$ be a set with\n\tuniform density $D(Q_j)$. Assume that \n\t\\begin{equation}\n\t\t\\label{EL.3.2}\n\t\t\\sum_{j\\geq 1} D(Q_j)< a.\n\t\\end{equation}\n\tThen one can find positive numbers $T_j$ \nand a non-zero Schwartz function $\\varphi$ on $\\R$ such that: \n\t\\begin{enumerate-math}\n\t\\item $\\spec(\\varphi)\\subset (0,a)$;\n\t\\item $\\varphi$ vanishes on the set $Q$ defined by\n\t\t\\begin{equation}\n\t\t\t\\label{EL.3.1}\n\t\t\tQ:= \\bigcup_{j\\geq 1} Q_j' , \\qquad\n\t\t\tQ_j':=Q_j\\setminus(-T_j,T_j).\n\t\t\\end{equation} \n\t\\end{enumerate-math}\n\\end{lem}\nBefore we prove \\lemref{EL.3}, let us first show how to deduce \\thmref{ET.1} from it.\n\\begin{proof}[Proof of \\thmref{ET.1}]\n\tLet $\\{x_j\\}$ $(j\\ge 1)$ be a sequence of points which are dense in $\\R^n$. For\n\teach $j$, choose an open ball $B_j$ centered at the point $x_j$, in such a way\n\tthat \n\t\\begin{equation}\n\t\t\\label{EL.4.1}\n\t\t\\sum_{j\\ge 1} \\mes(B_j)< \\frac{\\varepsilon}{\\det(\\Gamma)}.\n\t\\end{equation}\n\tConsider the sets $Q_j \\subset \\R$ defined by \n\t\\[ Q_j := \\left\\{ p_2(\\gamma^{*})\\; :\\; \\gamma^{*}\\in\\Gamma^{*}, \\;\n\t\tp_1(\\gamma^{*})\\in B_j \\right\\}.\n\t \\]\n\tThen each $Q_j$ is a model set, with uniform density \n\t\\[ D(Q_j)= \\det(\\Gamma) \\mes(B_j) \\]\n\taccording to \\lemref{EL.2}. Due to \\eqref{EL.4.1} this implies that \n\t\\[ \\sum_{j\\ge 1} D(Q_j)< \\varepsilon. \\]\n\t\\lemref{EL.3} therefore gives a sequence $\\{T_j\\}$ of positive numbers,\n\tand a non-zero Schwartz function $\\psi$ on $\\R$, \n\t$\\spec(\\psi) \\subset (0,\\varepsilon)$, such that $\\psi$ \n\tvanishes on the set $Q$ in \\eqref{EL.3.1}.\n\tHence $\\varphi := |\\psi|^2 \\geq 0$ is also a Schwartz function vanishing on\n\t$Q$, and $\\spec(\\varphi)\\subset (-\\varepsilon,\\varepsilon)$.\n\\par\n\tFor each $j\\ge 1$ there are only finitely many points \n\tof the lattice $\\Gamma^{*}$ lying in the set $B_j\\times (-T_j,T_j)$, so we\nmay choose an open ball $\\Omega_j$ contained in\n\t$B_j$ such that $\\Omega_j\\times (-T_j,T_j)$ has no points in common\n\twith $\\Gamma^{*}$. Notice that the set \n\t\\[ \\Omega := \\bigcup_{j\\ge 1} \\Omega_j \\]\n\tis an open, dense set in $\\R^n$.\n\\par\n\tWe claim that the spectrum of the measure\n\t\\eqref{RP.9.1} does not intersect the set $\\Omega$. Indeed, by \\eqref{RP.9.2}, an\n\telement of the spectrum is a point of the form $p_1(\\gamma^{*})$, where\n\t$\\gamma^{*}\\in \\Gamma^{*}$ and $\\varphi ( p_2(\\gamma^{*}) ) \\ne 0$. \n\tIf $p_1(\\gamma^{*})\\in \\Omega_j$ for some $j$, then we must have\n\t$|p_2(\\gamma^{*})|\\ge T_j$. Hence \n\\[ p_2(\\gamma^{*})\\in Q_j\\setminus(-T_j,T_j) \\subset Q,\n\\]\n which is not\n\tpossible as $\\varphi$ vanishes on $Q$. \n\\par\n\tWe conclude that the spectrum $S$ of the\n\tmeasure $\\mu$ is contained in the closed, nowhere dense set $\\R^n\\setminus\n\t\\Omega$. On the other hand, the support $\\Lambda$ of the\n\tmeasure is contained in the model set \n\t$\\Lambda (\\Gamma, (-\\varepsilon,\\varepsilon) )$, so this completes the proof. \n\\end{proof}\n\\subsection{ } \\label{section_8.4}\nIt remains to prove \\lemref{EL.3}. For this we will use\nthe celebrated Beurling and Malliavin theorem, see \\cite{bm67}.\n\\par\nFirst we recall the definition of the \\emph{Beurling-Malliavin upper\ndensity} (there are several equivalent ways to define this\ndensity). By a \\define{substantial system} of intervals we mean a\nsystem $\\{I_k\\}$ of disjoint open intervals on $\\R$, such that\n$\\inf_{k}|I_k|>0$, and \n\\[\t\\sum_{k}\\left( \\frac{|I_k|}{1+\\dist(0,I_k)} \n\t\\right)^2 = \\infty. \\]\n\\par\nIf $\\Lambda\\subset\\R$ is a discrete closed set, then its\nBeurling-Malliavin upper density $D^{*}(\\Lambda)$ is defined to be the\nsupremum of the numbers $d>0$, for which there exists a\nsubstantial system $\\{I_k\\}$ satisfying \n\\[\t\\frac{\\# (\\Lambda \\cap I_k)}{|I_k|}\\ge d \\]\nfor all $k$. If for any $d>0$ no such a system $\\{I_k\\}$ exists, then $D^{*}(\\Lambda)=0$.\n\\begin{thm}[Beurling and Malliavin \\cite{bm67}]\n\\label{ET.2}\nLet $\\Lambda\\subset \\R$ be a discrete closed set. Then for any\n$a>D^{*}(\\Lambda)$ one can find a non-zero function $\\varphi\\in\nL^{2}(\\R)$ such that: \n\\begin{enumerate-math}\n\\item $\\spec(\\varphi)\\subset (0,a)$;\n\\item $\\varphi$ vanishes on $\\Lambda$. \n\\end{enumerate-math}\n\\end{thm}\nBy multiplying $\\varphi$ by a Schwartz function with a sufficiently small \nspectrum, it is clear that one may assume the function $\\varphi$ in this \ntheorem to belong to the Schwartz class.\n\\par\n\\begin{remark}\nIt was also proved by \nBeurling and Malliavin that if $a0$, this would imply that \n\\[ \\sum_k \\left( \\frac{|I_k|}{1+\\dist(0,I_k)} \\right)^2 \\le \n\\sum_k \\big(1+\\dist (0,I_k) \\big)^{-4\/3} < \\infty,\\]\nwhich is not possible since the system $\\{I_k\\}$ is substantial.\n\\par\nNow consider an interval $I_k$ from the system, satisfying \\eqref{EL.3.8}. We will show\nthat \n\\begin{equation}\n\t\\label{EL.3.9}\n\t\\frac{\\# (Q\\cap I_k)}{|I_k|}0$, consider the measure\n\\[\\gamma_\\Lambda^R := (2R)^{-n} \\sum_{\\lambda,\\lambda'\\in \\Lambda\\cap [-R,R]^n}\\delta_{\\lambda'-\\lambda}. \\]\nIt is a finite measure on $\\R^n$ which is both positive and positive-definite.\n The uniform discreteness of $\\Lambda$ implies that the measures\n$\\gamma_\\Lambda^R$ are all translation-bounded, with the constant\nin \\eqref{tr-bdd-def} bounded uniformly with respect to $R$. \nHence there exists at least one\nweak limit point $\\gamma_\\Lambda$ of the measures\n$\\gamma_\\Lambda^R$ as $R\\to\\infty$. Any such limit point\n$\\gamma_\\Lambda$ is called an \\define{autocorrelation measure} of\nthe set $\\Lambda$.\n The measure $\\gamma_\\Lambda$ is also\n translation-bounded, positive and positive-definite. \n\\par\nThe positive-definiteness of $\\gamma_\\Lambda$ implies that\n its Fourier transform\n$\\hat{\\gamma}_\\Lambda$ is a positive measure. It is called a\n\\define{diffraction measure} of $\\Lambda$. If the measure\n$\\hat{\\gamma}_\\Lambda$ is purely atomic, then \nits support $S$ is called a\n\\define{diffraction spectrum} of $\\Lambda$, and $\\Lambda$ is\nsaid to be a \\define{pure point diffractive set}.\n\\par\nMore generally, one can define diffraction by any \ntranslation-bounded measure $\\mu$ on\n$\\R^n$, in a similar way. Denote by $\\mu_R$ the restriction of $\\mu$ to the\ncube $[-R,R]^n$, and define a measure $\\tilde{\\mu}_R$ by \n\\[ \\tilde{\\mu}_R(E):= \\overline{\\mu_R(-E)}.\\]\nThen the measures \n\\[\\gamma_\\mu^R:=(2R)^{-n} \\; \\mu_R * \\tilde{\\mu}_R\\]\nare uniformly translation-bounded and so \nhave at least one weak limit point $\\gamma_\\mu$ as $R\\to\\infty$, and any such\nlimit point is called an autocorrelation measure\nof $\\mu$. It is again a translation-bounded, positive-definite\nmeasure, and if $\\mu$ is a positive measure then\nalso $\\gam_\\mu$ is positive.\nThe diffraction measure\n$\\hat{\\gamma}_\\mu$ and the diffraction spectrum $S$\n(assuming that $\\hat{\\gamma}_\\mu$ is purely atomic) are also\ndefined in a similar way. \n\\par\nNotice that the diffraction by a Delone set $\\Lam$ described above, is included as a\nspecial case which corresponds to diffraction by the measure \n\\begin{equation}\n\t\\label{DD.1}\n\t\\mu = \\sum_{\\lambda\\in\\Lambda}\\delta_\\lambda.\n\\end{equation}\n\n\\subsection{} \\label{9.2}\nA Delone set $\\Lambda \\subset \\R^n$ is said to be of\n\\define{finite local complexity} if for every $R>0$ there\nare only finitely many different sets of the form\n\\[ (\\Lambda - \\lambda)\\cap B_R, \\quad \\lambda \\in \\Lambda. \\] \nIt is easy to verify that this condition is equivalent to the requirement that \n$\\Lambda-\\Lambda$ is a discrete closed set. \n\\par\nNotice that if $\\mu$ is a translation-bounded measure supported by a Delone set $\\Lambda$ of\nfinite local complexity, then any autocorrelation measure $\\gamma_\\mu$ of $\\mu$ must be supported by \nthe set $\\Lambda-\\Lambda$. In particular, $\\gamma_\\mu$ is a discrete measure.\n\\par\nModel sets are well-studied\nexamples of non-periodic Delone sets of finite local complexity,\nwith pure point diffraction in Hof's sense. More precisely, if\n$\\Lambda=\\Lambda(\\Gamma,\\Omega)$ is a model set defined by\n\\eqref{RP.9.0} and such that the\nboundary of the ``window'' $\\Omega$ is a set of Lebesgue measure\nzero, then $\\Lambda$ is a pure point diffractive set, with a\ndense countable diffraction spectrum (see for example \\cite[Section 9.4]{bagr}).\n\n\n\n\\subsection{} \nLet $\\Lambda \\subset \\R^n$ be a Delone set of finite local complexity.\nAssume that the diffraction spectrum $S$ is uniformly discrete. \nIs it true that $S$ must have a periodic structure?\n\\par\nThe question was raised in \\cite[Problem 4.2(a)]{lag2}.\nIt follows from our previous results that the answer is positive:\n\\begin{thm}\n\t\\label{DT.1}\nSuppose that\n\\begin{enumerate-math}\n\\item\n$\\Lambda \\subset \\R^n$ is a\n\tDelone set of finite local complexity;\n\\item\n$\\mu$ is a positive, translation-bounded measure supported by $\\Lam$;\n\\item\n$\\gamma_\\mu$ is an autocorrelation measure of $\\mu$; and\n\\item\nthe support $S$ of the diffraction\n\tmeasure $\\hat{\\gamma}_\\mu$ is a uniformly discrete set. \n\\end{enumerate-math}\nThen $S$ is contained in a finite union of translates of a certain lattice,\nand the diffraction measure has the form \\eqref{RM.3}.\n\\end{thm}\n\n\n\\begin{proof}\nThe \n\tautocorrelation measure $\\gamma_\\mu$ is positive, and is supported by $\\Lambda-\\Lambda$. Hence the\n\tdiffraction measure $\\hat{\\gamma}_\\mu$ is a positive-definite measure on $\\R^n$, whose\n\tsupport $S$ is a uniformly discrete set, and whose spectrum is contained in the\n\tdiscrete closed set $\\Lambda-\\Lambda$. \\thmref{RT.5} (applied to the measure\n\t$\\hat{\\gamma}_\\mu$) therefore yields that $\\hat{\\gamma}_\\mu$\n\tis of the form \\eqref{RM.3}. As a consequence, $S$ \n\tmust be contained in a finite union of translates of a lattice.\n\\end{proof}\n\n\n\nIn particular \\thmref{DT.1} applies to the measure \\eqref{DD.1}.\nIn this case the result shows that if a Delone set\n$\\Lambda$ of finite local complexity is pure point diffractive, \nand if the diffraction spectrum $S$ is uniformly discrete, then\n$S$ is contained in a finite union of translates of a lattice,\nand the diffraction measure has the form \\eqref{RM.3}.\nSo we obtain \\thmref{RT.8}. \n\n\n\n\t\\subsection{}\nIf $\\Lambda$ is a Meyer set, then the conclusion in the previous result\nremains true even if the spectrum is just a discrete closed set, and\nwithout the positivity of the measure:\n\\begin{thm}\n\t\\label{DT.2}\nSuppose that\n\\begin{enumerate-math}\n\\item\n$\\Lambda$ is a Meyer set in $\\R^n$;\n\\item\n$\\mu$ is a translation-bounded measure supported by $\\Lam$;\n\\item\n$\\gamma_\\mu$ is an autocorrelation measure of $\\mu$; and\n\\item\nthe support $S$ of the diffraction\n\tmeasure $\\hat{\\gamma}_\\mu$ is a discrete closed set. \n\\end{enumerate-math}\nThen the same conclusion as in \\thmref{DT.1} is true.\n\\end{thm}\n\nThis can be deduced from either Theorem \\ref{RT.5} or \\ref{RT.7},\nusing the fact that the auto\\-correlation measure $\\gamma_\\mu$ is a positive-definite measure,\nsupported by the Meyer set $\\Lambda-\\Lambda$. \n\n\n\\subsection*{Remarks} \n{\\bf 1.}\\;\\;Similarly, one can prove a dichotomy result for the diffraction spectrum\nof a measure $\\mu$ supported by a Meyer set: either\nthe spectrum\nis uniformly discrete, or it has a relatively dense set of accumulation points\n(using Theorem \\ref{AT.5} or \\ref{CT.1}).\n\\par\n{\\bf 2.}\\;\\;In the latter case, the spectrum $S$ need not be dense in any ball.\nOne can verify that the measure constructed in the proof of \\thmref{ET.1}\nis an autocorrelation of another measure whose support is also a Meyer set.\n\n\n\\section{Non pure point spectrum}\n\n\\subsection{}\nIn crystallography it is often interesting to consider also discrete measures\n$\\mu$, whose Fourier transform $\\hat{\\mu}$ is a measure which has both a pure point\ncomponent and a continuous one. The pure point component is often referred to as ``Bragg\npeaks'', while the continuous component is called ``diffuse background''.\n\\par\n Let $\\mu$ be a\n(slowly increasing) measure on $\\R^n$ with discrete support $\\Lambda$: \n\\begin{equation}\n\t\\label{FE.1}\n\t\\mu = \\sum_{\\lambda\\in \\Lambda} \\mu(\\lambda)\\delta_\\lambda, \\quad \\mu(\\lambda)\\ne 0.\n\\end{equation}\nAssume that $\\hat{\\mu}$ is also a slowly increasing measure, and consider its decomposition \n\\begin{equation}\n\t\\label{FE.2.2}\n \\hat{\\mu} = \\hat{\\mu}_d+\\hat{\\mu}_c \n\\end{equation}\ninto a sum of a pure point measure\n\\begin{equation}\n\t\\label{FE.2}\n\t\\hat{\\mu}_d = \\sum_{s\\in S}\\hat{\\mu}_d(s)\\delta_s, \\quad \\hat{\\mu}_d(s)\\ne 0,\n\\end{equation}\nand a continuous measure $\\hat{\\mu}_c$. The set $S$ is the support of the discrete part $\\hat{\\mu}_d$.\n\\par\nWe can extend our previous results to this more general situation,\nusing the following result (\\thmref{RT.10}): If\n the support $\\Lambda$ is uniformly discrete, then $\\hat{\\mu}_d$ \n is the Fourier transform of another measure\n\t$\\mu'$, whose support $\\Lambda'$ is also a uniformly discrete set.\n\n\n\n\\subsection*{Remark} \nWe will see from the proof that the new measure $\\mu'$ is a weak limit of \ntranslates of $\\mu$. Hence, in particular, the following is true:\n\\begin{enumerate-math}\n\\item \\label{FT.1.i} If $\\mu$ is a positive measure, then also $\\mu'$ is positive;\n\\item \\label{FT.1.ii}\n If $\\Lambda-\\Lambda$ is a discrete closed set, \n\tthen also $\\Lambda'-\\Lambda'$ is a discrete closed set; \n\\item \\label{FT.1.iii}\n If $\\Lambda-\\Lambda$ is uniformly discrete,\n\tthen also $\\Lambda'-\\Lambda'$ is uniformly discrete.\n\\end{enumerate-math}\nProperty \\ref{FT.1.i} is obvious.\nProperties \\ref{FT.1.ii} and \\ref{FT.1.iii} follow from the\n fact that $\\Lambda'-\\Lambda'$ must be contained in the closure of\nthe set $\\Lambda-\\Lambda$.\n\n\n\n\\subsection{}\nFirst we give the proof of \\thmref{RT.10}. We will use the following lemmas:\n\\begin{lem}\n\t\\label{FL.1}\n\tLet $\\nu$ be a finite measure on $\\R^n$. Then \n\t\\[ \\lim_{R\\to\\infty} (2R)^{-n}\\int_{[-R,R]^n}\\left| \\hat{\\nu}(t)\\right|^2dt\n\t= \\sum_{a}\\left|\\nu (\\{a\\} )\\right|^2, \\]\n\twhere $a$ goes through all the atoms of the measure $\\nu$. \n\\end{lem} \nThis is the well-known Wiener's lemma in $\\R^n$. \n\\begin{lem}\n\t\\label{FL.2}\n\tLet $\\nu$ be a (slowly increasing) continuous measure on $\\R^n$. Then there exist\n\tvectors $\\omega_k\\in \\R^n$ $(k\\ge 1)$ such that the measures \n\t\\begin{equation}\n\t\t\\label{FL.2.1}\n\t\t\\nu_k(x):=e^{-2\\pi i\\dotprod{\\omega_k}{x}} \\,\\nu(x)\n\t\\end{equation}\n\ttend to zero as $k\\to\\infty$ in the space of temperate distributions. \n\\end{lem}\n\\begin{proof}\n\tLet $\\{\\varphi_j\\}$, $j\\geq 1$, be a sequence of functions dense in the Schwartz\n\tspace. Define \n\t\\[ \\Phi_k(t) := \\sum_{j=1}^{k}\\left| \\int \\varphi_j(x)e^{-2\\pi i\n\t\\dotprod{t}{x}}d\\nu(x)\\right|^2, \\quad t\\in \\R^n. \\]\n\tFor each $j$, the measure $\\varphi_j\\cdot \\nu$ is finite and continuous, hence\n\tby \\lemref{FL.1} we have\n\t\\[ \\lim_{R\\to\\infty}(2R)^{-n}\\int_{[-R,R]^n} \\Phi_k(t)dt = 0. \\]\n\tThis implies that for each $k$ one can find $\\omega_k\\in\\R^n$ such that $\\Phi_k(\\omega_k)<\n\t1\/k^2$.\n\\par\n Now consider the measure $\\nu_k$ defined by \\eqref{FL.2.1}. We have\n\t\\[ \\left|\\dotprod{\\nu_k}{\\varphi_j}\\right| < \\frac{1}{k} \\quad (k\\ge j) \\]\n\tand hence $\\dotprod{\\nu_k}{\\varphi_j}\\to 0$ as $k\\to\\infty$, for each $j$.\nSince $\\nu$ is a slowly increasing measure, the sequence $\\nu_k$ is uniformly bounded\n in the space of temperate distributions. Since the $\\varphi_j$ are dense in the\nSchwartz space, we can conclude that $\\dotprod{\\nu_k}{\\varphi}\\to 0$ as $k\\to\\infty$,\nfor every Schwartz function $\\varphi$. This proves the lemma.\n\\end{proof}\n\\begin{proof}[Proof of \\thmref{RT.10}]\n\tUse \\lemref{FL.2} to find vectors $\\omega_k$ such that \n\t\\[ e^{-2\\pi i \\dotprod{\\omega_k}{x}} \\,\\hat{\\mu}_c(x)\\to 0, \\quad k\\to \\infty \\]\n\tin the space of temperate distributions. By taking a subsequence if necessary we\n\tcan also assume that the sequence $e^{2\\pi i \\dotprod{\\omega_k}{s}}$ has a limit as\n\t$k\\to\\infty$, for each $s\\in S$. Let $\\{k_j\\}$ be a sufficiently fast increasing sequence\n\tsuch that \n\t\\begin{equation}\n\t\t\\label{FT.1.0}\n\t e^{2\\pi i \\dotprod{\\omega_j-\\omega_{k_j}}{x}} \\, \\hat{\\mu}_c(x)\\to 0, \\quad\n\tj\\to\\infty. \n\t\\end{equation}\n\tSince the exponential in \\eqref{FT.1.0} tends to $1$ on $S$, \tit follows that the measure\n\t\\begin{equation}\n\t\t\\label{FT.1.1}\n\t\te^{2\\pi i \\dotprod{\\omega_j-\\omega_{k_j}}{x}} \\,\\hat{\\mu}(x) \n\t\\end{equation}\n\ttends to $\\hat{\\mu}_d$ as $j\\to \\infty$. \nThe measure in \\eqref{FT.1.1} is the\n\tFourier transform of the measure \n\t\\[ \\mu_j(t):= \\mu(t+\\omega_j-\\omega_{k_j}). \\]\n\tTherefore, $\\mu_j$ tends as $j\\to\\infty$ to a certain distribution $\\mu'$, whose\n\tFourier transform is $\\hat{\\mu}_d$. By \\lemref{AL.4} the measure $\\mu$ is\n\ttranslation-bounded, and therefore also $\\mu'$ is a translation-bounded measure.\n\tThe measure $\\mu_j$ is supported by the set\n\t\\[ \\Lambda_j := \\Lambda-\\omega_j+\\omega_{k_j}, \\]\n\twhich is uniformly discrete with $d(\\Lambda_j)=d(\\Lambda)$. Hence the support $\\Lambda'$\n\tof the measure $\\mu'$ must satisfy $d(\\Lambda')\\ge d(\\Lambda)$, so $\\Lam'$ is also a uniformly\n\tdiscrete set. \n\\end{proof}\n\\subsection{}\n\\thmref{RT.10} allows to extend our previous results to the case when the measure\n$\\hat{\\mu}$ has also a continuous component. For example, we have:\n\\begin{thm}\n\t\\label{FT.2}\n\tLet $\\mu$ be a measure on $\\R^n$ satisfying \\eqref{FE.1}--\\eqref{FE.2}. Assume\n\tthat $\\Lambda$ is uniformly discrete, $S$ is discrete and closed, and at least one\n\tof the following additional conditions is satisfied: \n\t\\begin{enumerate-math}\n\t\\item \\label{FT.2.i} $\\mu$ is positive, and $S$ is uniformly discrete; \n\t\\item \\label{FT.2.ii} $\\hat{\\mu}_d$ is positive; \n\t\\item \\label{FT.2.iiii} $n=1$, and $S$ satisfies condition \\eqref{RM.4};\n\t\\item \\label{FT.2.iv} $\\Lambda-\\Lambda$ is uniformly discrete. \n\t\\end{enumerate-math}\n\tThen $S$ is a uniformly discrete set, contained in a finite union of translates of\n\ta lattice, and the measure $\\hat{\\mu}_d$ has the form \\eqref{RM.3}.\n\\end{thm}\nTo prove this we consider the measure $\\mu'$ given by \\thmref{RT.10}, and apply to this\nmeasure one of Theorems \\ref{RT.0}, \\ref{RT.5}, \\ref{RT.6} or \\ref{RT.7}, according to which\none of the conditions \\ref{FT.2.i}--\\ref{FT.2.iv} in \\thmref{FT.2} is satisfied. \n\\par\nIn a similar way, one can extend the dichotomy results given in Theorems \\ref{AT.5} or \\ref{CT.1}.\n\\subsection{}\nThe same applies to Hof's diffraction by measures supported on Meyer sets:\n\\begin{thm}\n\t\\label{FT.3}\n\tLet $\\mu$ be a translation-bounded measure on $\\R^n$, \n\tsupported by a Meyer set $\\Lambda$. Let $\\gamma_\\mu$ be an autocorrelation measure of\n\t$\\mu$, and denote by $S$ the support of the discrete part of the diffraction measure $\\hat{\\gamma}_\\mu$. Then, either\n\t\\begin{enumerate-math}\n\t\\item $S$ is a uniformly discrete set, contained in a finite union of translates\n\t\tof a lattice, and the discrete part of $\\hat{\\gamma}_\\mu$ has the\n\t\tform \\eqref{RM.3}; or \n\t\\item $S$ is not a discrete closed set, and moreover the set of accumulation\n\t\tpoints of $S$ is relatively dense. \n\t\\end{enumerate-math}\n\\end{thm}\nTo prove this one can first apply \\thmref{RT.10} to the measure\n$\\gamma_\\mu$ which is supported by the Meyer set $\\Lam-\\Lam$,\nand then use Theorem \\ref{RT.0} and Theorem \\ref{AT.5} or \\ref{CT.1}.\n\n\\subsection*{Remark} \nAn alternative proof of \\thmref{FT.3}, which does not rely\non \\thmref{RT.10}, can be given using the following:\n\\begin{lem}\n\t\\label{FL.4}\n\tLet $\\nu$ be a translation-bounded measure on $\\R^n$, and assume that\n$\\ft\\nu$ is a slowly increasing measure. Then $\\nu$ has a unique autocorrelation measure\n$\\gamma_\\nu$, and the diffraction measure $\\hat{\\gamma}_\\nu$ is a pure point\nmeasure given by\n\\[\n\\ft{\\gamma}_\\nu \n\t= \\sum_{a}\\left|\\ft\\nu (\\{a\\} )\\right|^2 \\delta_a, \\]\n\twhere $a$ goes through all the atoms of the measure $\\ft\\nu$. \n\\end{lem}\nTo prove \\thmref{FT.3} using this lemma, let $\\nu := \\gamma_\\mu$.\nThen the autocorrelation measure $\\gamma_\\nu$ is a discrete measure, supported by the Meyer \nset $\\Lam+\\Lam-\\Lam-\\Lam$, and \nby \\lemref{FL.4} the measure $\\hat{\\gamma}_\\nu$\nis a pure point measure, with the same support as the discrete\npart of $\\ft{\\gam}_\\mu$. So our previous results can be applied\nto the measure $\\gamma_\\nu$.\n\\par\nWe omit the proof of \\lemref{FL.4}.\n\n\n\n\n\\section{Remarks. Open problems}\n\n\\subsection{} \n Very recently, Y.~Meyer has found \\cite{mey4} an interesting version of \n\\thmref{RT.4}. Namely, he constructed measures $\\mu$\n whose supports and spectra are\n discrete closed sets, which can be described by simple effective formulas. \n He also proved that the parameters of this construction can be chosen so \nthat both $\\Lam$ and $S$ are rationally independent sets. This is a stronger ``non-periodicity''\n condition than in \\thmref{RT.4}. However, such a measure cannot be translation-bounded,\n see \\cite[Lemma 5]{mey4}.\n \n It should be mentioned that the last paper contains some other examples of\n measures with discrete closed supports and spectra. See also \\cite{kol}.\n All these examples, in one way or another, are based on the classical Poisson\n summation formula. \nQuestion: can one construct an example which in no way is based\non Poisson's formula?\n\n\n\\subsection{}\nWe mention some problems which are left open. \n\\par\n1.~The first one concerns the positivity assumption in\n\\thmref{RT.0} in several dimensions.\nLet $\\mu$ be a measure on $\\R^n$, $n>1$, \nsatisfying \\eqref{RM.1} and \\eqref{RM.2}. Assume that $\\Lambda, S$ are both\nuniformly discrete sets. Is it true that $\\Lambda$ can be covered\nby a finite union of translates of several, not necessarily\ncommensurate, lattices?\n(an example in \\cite{fav} shows that $\\Lam$ need not be\ncontained in a finite union of translates of a \\emph{single} lattice).\n\\par\n2.~A second problem concerns the positive-definiteness assumption\nin \\thmref{RT.5}, even in dimension one:\nLet $\\mu$ be a measure on $\\R$, with uniformly discrete support\n$\\Lam$ and discrete closed spectrum $S$. Does it follow that\n$S$ must be also uniformly discrete?\n\\par\n3.~The following question is also open: can one get a \\define{positive} measure in \\thmref{RT.4}\\,?\n\n\n\\subsection{}\nOur approach to prove \\thmref{RT.0} (see \\cite{lo2}) involved a combination\nof analytic and discrete combinatorial considerations. \nIn the latter part, a conclusion about the arithmetic structure\nof a set $\\Lambda\\subset \\R^n$ was derived from information on\ndiscreteness of $\\Lambda-\\Lambda$. In that point we relied on results\nwhich go back to Meyer \\cite{mey1}. \n\\par\nIn this context, it is worth to mention Freiman's theorem \\cite{freim},\nwhich states that a finite set\n$A$ such that $\\#(A+A)\\le K \\# A$, must be contained in a\n``generalized arithmetic progression'' whose dimension and size\nare controlled in terms of the constant $K$. It might be\ninteresting to see whether it can also be used for a \nproof of \\thmref{RT.0}.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\n\n\\subsection{Pattern-forming systems}\nPeriodic, stripe-like patterns emerge in a self-organized fashion in a variety of experiments, ranging from Rayleigh-B\\'enard convection to open chemical reactors. Such regular, periodic patterns are usually studied in domains with idealized periodic boundary conditions, where existence and stability can be readily obtained using classical methods of bifurcation theory. The simplest example for such pattern-forming systems is the Swift-Hohenberg equation\n\\[\nu_t=-(\\Delta +1)^2 u +\\delta^2 u - u^3,\\qquad (x,y) \\in \\mathbb{R}^2,\n\\]\n\nwhich is known to possess periodic patterns of the form \n\\[\nu(x,y)=u_*(kx;k), u_*(\\xi+2\\pi;k)=u_*(\\xi;k),\n\\]\nwith $k\\sim 1$,\nfor small $\\delta$. \n\nBeyond periodic boundary conditions, the dynamics for $\\delta\\sim 0$ can be approximated using the amplitude equation formalism. In the case of the (isotropic) Swift-Hohenberg equation, one finds the Newell-Whitehead-Segel equation\n\\[\nA_T=-(\\partial_X-\\mathrm{i}\\partial_{YY})^2 A + A-A|A|^2,\n\\]\nusing an Ansatz\n\\[\nu(x,y,t)=\\delta A(\\delta x,\\delta y, \\delta^2 t)\\mathrm{e}^{\\mathrm{i} x}+c.c.,\n\\]\n\nand expanding to order $\\delta^3$, as a solvability condition \\cite{cross1993pattern,mielke2002ginzburg}. There are several difficulties with the NWS equations and their validity as an approximation \\cite{schneider1995validity}, related to the fact that the original equation is isotropic, while the expansion singles out a preferred wave vector, here the vector $k=(1,0)^T$. The situation is simplified in anisotropic pattern-forming systems such as \n\\[\nu_t = -(\\Delta +1)^2u+\\partial_{yy}u +\\delta^2 u - u^3,\n\\]\nwhere a similar Ansatz leads to the isotropic (sic!) Ginzburg-Landau equation\n\\begin{equation}\\label{GL}\nA_T=\\Delta A + A-A|A|^2,\n\\end{equation}\npossibly after rescaling $X$ and $Y$. \n\nMore drastically, one can approximate the dynamics near periodic patterns using an Ansatz\n\\[\nu(t,x,y)=u_*(\\Phi(\\delta x,\\delta y,\\delta^2 t);|\\nabla \\Phi|),\n\\]\nwhere one obtains as a compatibility condition the phase-diffusion equation \n\\[\n\\Phi_T=\\Delta_{X,Y}\\Phi,\n\\]\nfor suitable values of the wavenumber $|\\nabla \\Phi|$, after possibly rescaling $X,Y$. \n\nBoth, amplitude and phase-diffusion equations can be shown to be good approximations under suitable choices of initial conditions, on time scales $T=\\mathrm{O}(1)$; see for instance \\cite{doelman2009dynamics,mielke2002ginzburg} and references therein.\n\n\\subsection{Inhomogeneities}\n\nLocal impurities in experiments sometimes have minor, sometimes more dramatic effects on the resulting patterns. It is known, for instance, that target-like patterns can nucleate at impurities in the Belousov-Zhabotinsky reaction; see \\cite{kollar2007coherent} for an analysis in this direction. Also, spiral wave anchoring at impurities such as arteries can have dramatic impact on excitable media \\cite{munuzuri1998attraction}. Effects in Swift-Hohenberg-like systems appear to be more subtle and are the main focus of our present study. We focus on the somewhat simpler case of the isotropic GL equation \\eqref{GL}, modeling anisotropic pattern-forming systems, with an added localized inhomogeneity,\n\\begin{equation}\\label{e:cgli}\nA_T=\\Delta A + A-A|A|^2+\\varepsilon g(x,y).\n\\end{equation}\n\nWe intend to study inhomogeneities in the isotropic SH equation \n\\[\nu_t=-(\\Delta +1)^2u+\\delta^2 u - u^3+\\varepsilon g(x,y),\n\\]\nin future work. \n\nIn order to illustrate the difficulties that arise, consider the more dramatic simplification of the phase-diffusion approximation with an inhomogeneity \\footnote{We stress however that one cannot derive such an approximation in the presence of inhomogeneities due to the different scalings of $\\Phi$ and $g$.},\n\\[\n\\Phi_t=\\Delta \\Phi+\\varepsilon g(x,y).\n\\]\nStationary patterns here are solutions to the Poisson equation $\\Delta\\Phi=-\\varepsilon g(x,y)$, with solutions exhibiting logarithmic growth at infinity. Thinking of the phase-diffusion as an approximation to a larger system, one would like a robust way of solving the stationary equation, if possible relying on an implicit function theorem. The difficulty here is that the Laplacian is not invertible on $L^2$, say, not even Fredholm as an unbounded operator. \n\nA remedy, in this context, are spaces with algebraic weights, which we refer to as Kondratiev spaces. To be precise, define $\\langle \\mathbf{x}\\rangle=\\sqrt{x^2+y^2+1}$ and $M^{2,p}_{\\gamma}$ as the completion of $C_0^{\\infty}$ in the norm\n\\[\n\\|u\\|_{M^{2,p}_{\\gamma}}:=\\sum_{j+k= 2}\\|\\langle {\\bf x} \\rangle^{\\gamma+2}\\partial^j_x\\partial^k_yu \\|_{L^p}\n+\n\\sum_{j+k= 1}\\| \\langle {\\bf x} \\rangle^{\\gamma+1}\\partial^j_x\\partial^k_yu\\|_{L^p}\n+\\| \\langle {\\bf x} \\rangle^{\\gamma} u\\|_{L^p}.\n\\]\nNote that the algebraic weights increase with the number of derivatives, making the different parts of the norms scale in the same fashion, as opposed to norms in the classical Sobolev spaces $W^{2,p}_{\\gamma}$, with norm \n\\[\n\\|u\\|_{W^{2,p}_\\gamma}:=\\sum_{j+k\\leq 2}\\|\\langle {\\bf x} \\rangle^{\\gamma} \\partial^j_x\\partial^k_yu \\|_{L^p}.\n\\]\nIt turns out that the Laplacian is Fredholm for suitable $\\gamma$ on $M^{2,p}_{\\gamma}$ \\cite{McOwen}, albeit with negative index in dimension 2 when $\\gamma>0$. This negative index makes ``explicit'' far-field corrections via logarithmic terms, just as seen in the Green's function of the Laplacian, necessary when describing the far-field effect of localized inhomogeneities on periodic patterns. This result also holds for a certain class of elliptic operators with coefficients that decay sufficiently fast at infinity \\cite{lockhart1981fredholm}, but we are not aware of results for elliptic operators without scaling invariance. This represents a difficulty when looking at the linearization of \\eqref{e:cgli}, which for $k=0$ decouples into $\\Delta$ which is, and $\\Delta -I$, which is not scale invariant. For $k \\neq 0$ these components couple and simple scale invariance is lost. Furthermore, the linearization is in general not a small or compact perturbation of a scale invariant operator.\n\n\n\n\n\n\n\n\\subsection{Main results}\n\nTo state our main results, we consider the stationary solutions of \\eqref{e:cgli},\n\\begin{equation}\\label{stationaryGL}\n0=\\Delta A + A - A|A|^2+\\varepsilon g(x,y).\n\\end{equation}\nFor $\\varepsilon=0$, the system possesses ``stripe patterns'' \n\\[\nA(x,y)=\\sqrt{1-k^2}\\mathrm{e}^{\\mathrm{i} k x},\n\\]\nfor wavenumbers $|k|<1$. Those solutions are linearly stable for $|k|<1\/\\sqrt{3}$ and unstable for $|k|>1\/\\sqrt{3}$. The instability mechanism is known as the Eckhaus (sideband) instability. We are now ready to state our main result.\n\n\\begin{thm}\\label{MainTh1}\n\nFix $k$ with $|k|<1\/\\sqrt{3}$ and suppose that $g\\in W^{2,2}_{\\beta}$ for some $\\beta>2$. Then there exists a family of solutions to \\eqref{stationaryGL},\n\\[\nA(x,y;\\varepsilon,\\varphi)=S(x,y;\\varepsilon,\\varphi)\\mathrm{e}^{\\mathrm{i}\\Phi(x,y;\\varepsilon,\\varphi)},\n\\]\nwith $A(x,y;0,\\varphi)=\\sqrt{1-k^2}\\mathrm{e}^{\\mathrm{i} (k x+\\varphi)}$. Moreover, $A(x,y;\\varepsilon,\\varphi)$ is smooth in all variables and satisfies the following expansions in $x,y$ for fixed $\\varepsilon,\\varphi$,\n\\begin{align}\nS(x,y;\\varepsilon,\\varphi)&\\to \\sqrt{1-k^2}\\\\\n\\Phi(x,y;\\varepsilon,\\varphi)-kx-\\frac{c(\\varepsilon,\\varphi)}{ 2k \\sqrt{1-k^2}} \\log(\\alpha x^2+y^2)&\\to \\Phi_\\infty(\\varepsilon)+\\varphi,\n\\end{align}\nfor $|\\mathbf{x}|\\to\\infty$, with $\\alpha=\\frac{1-k^2}{1-3k^2}$, for some smooth function $c(\\varepsilon,\\varphi)$ with expansion\n\\[\nc(\\varepsilon,\\varphi)=\\varepsilon c_1(\\varphi)+\\mathrm{O}(\\varepsilon^2),\n\\]\nwhere\n\\[\nc_1(\\varphi)=\\frac{\\sqrt{1-3k^2}}{\\pi(1-k^2)} \\iint \\Imag[g(x,y)\\mathrm{e}^{-\\mathrm{i}(kx+\\varphi)}] \\mathrm{d} x \\mathrm{d} y.\n\\]\n\n\\end{thm}\n\n\n\\begin{rem}\n\n\\begin{enumerate}\n\\item Our approach gives more detailed expansions than stated. In fact, we obtain a decomposition of $S$ and $\\Phi$ into a localized part that is smooth in $\\varepsilon$, uniformly in $\\mathbf{x}$, and an explicit logarithmic far-field correction with coefficient $c(\\varepsilon,\\varphi)$; see the Ansatz \\eqref{Ansatz}. Also, in the class of functions with this particular form, the solutions described in the theorem are locally unique. \n\n\\item The expression for $c_1(\\varepsilon,\\varphi)$ is reminiscent of a projection onto the kernel. Indeed, the integral represents the scalar product $(u,v)=\\Real \\int u\\bar{v}$ of the perturbation $\\varepsilon g$ and the ``kernel'' of the linearization induced by the infinitesimal phase rotation, $\\frac{\\mathrm{d}}{\\mathrm{d}\\varphi}\\mathrm{e}^{\\mathrm{i} {kx+\\varphi}}$. Vanishing of such a scalar product indicates persistence of solutions in problems where the linearization is Fredholm, such as in the Melnikov analysis for homoclinic bump-like solutions. \n\nIn fact, $c_1$ possesses at least 2 zeroes. Assuming that $c_1(\\varphi_*)=0$, $c_1'(\\varphi_*)\\neq 0$, we can find $\\varphi_*(\\varepsilon)$ so that\n\\[\nc(\\varepsilon,\\varphi_*(\\varepsilon))=0.\n\\]\nInspection of the expansion in the theorem shows that $\\varphi$ is the phase shift of the underlying pattern in the far field. For these specific values of $\\varphi$, the correction in the far field to the periodic pattern is bounded and small, while for other values of $\\varphi$ the correction is unbounded in the phase. We interpret this result by referring to $\\varphi_*(\\varepsilon)$ as the \\emph{selected phase}. In other words, introducing inhomogeneities induces a selected phase shift of the primary pattern which accommodates stationary solutions without logarithmic corrections. Numerical simulations confirm this phenomenon, with a diffusive spread of the phase shift in the domain. It would be interesting to establish this diffusive convergence to a selected phase analytically.\n\n\\item Very similar results hold in space dimensions 1 and 3. In one space dimension, the analysis is easier since ODE methods can be used to analyze stationary solutions. In fact, the analysis reduces to a Melnikov analysis for the intersection of center-stable and center-unstable manifolds. In this one-dimensional context, the analysis also immediately carries over to the Swift-Hohenberg equation. On the other hand, the 3-dimensional case is easier than the two-dimensional case considered here since the corrections needed to compensate for negative Fredholm indices are decaying like $1\/|\\mathbf{x}|$. In fact, the Laplacian is invertible for suitable weights $\\gamma$ in Kondratiev spaces in $\\mathbb{R}^3$. \n\n\n\\end{enumerate}\n\n\\end{rem}\n\n\nThe structure of this paper is as follows. In Section \\ref{weightedspaces} we give a more detailed description of weighted Sobolev spaces and Kondratiev spaces, and state Fredholm properties of the Laplacian in this setting. Next, in Section \\ref{summary}, we summarize the procedure leading up to the main result, first explaining the difficulties encountered when analyzing the linearization of equation \\eqref{stationaryGL} about stripe patterns, and then describing the linear operator $T$ that we use to overcome these difficulties. In Section \\ref{sectionlinear} we use the results from Section \\ref{weightedspaces} to explore the Fredholm properties of this last operator and show that by adding logarithmic corrections we can obtain an invertible operator $\\hat{T}$. Section \\ref{sectionnonlinear} establishes properties of the nonlinearity in our functional analytic setting. We show that the full operator $\\hat{F}$ is well defined, continuously differentiable, with invertible linearization\n$\\hat{T}$. Finally, in Section \\ref{Mainresult}, we prove our main result using the Implicit Function Theorem. \n\n\n\\section{Fredholm properties of the linearization and weighted spaces} \\label{weightedspaces}\n\nThis section is intended as a summary of theorems and results that describe weighted spaces and their properties. These results are the basis for the analysis in the following sections and will allow us to conclude Fredholm properties of the linearized operators considered therein.\n\n\n\\subsection{Weighted Sobolev spaces}\nThroughout the paper we will use the symbol $W^{s,p}_{\\gamma}$ to denote weighted Sobolev spaces, which we define as the completion of $C^{\\infty}_0(\\mathbb{R}^n)$ under the norm\n\\[ \\| u\\|_{W^{s,p}_{\\gamma}} = \\sum_{|\\alpha|\\leq s} \\| \\langle {\\bf x} \\rangle^{\\gamma} D^{\\alpha} u \\|_{L^p}.\\]\n\nHere, $\\langle {\\bf x} \\rangle = ( 1 + |{\\bf x}|^2)^{1\/2}$, ${\\bf x}=(x,y)$, $\\gamma \\in \\mathbb{R}$, and $s$ is a positive integer. \n\nWe start with a generalization of the invertibility of the operator $\\Delta- I$ to weighted spaces.\n\n \n\\begin{prop}\\label{Sobolev}\nThe operator $\\Delta -a : W_{\\gamma}^{2,p} \\rightarrow L^p_{\\gamma}$ is invertible for all real numbers $a >0$ and $p \\in [1, \\infty)$. \n\\end{prop}\n\n\n\n\\begin{lem}\\label{notFredholm}\n The operator $\\Delta_{\\gamma}:W^{2,p}_{\\gamma} \\rightarrow L^p_{\\gamma}$ is not a Fredholm operator for $p \\in [1, \\infty)$.\n\\end{lem}\n\n\nBoth results follow in a straightforward fashion by noticing that $L^2_\\gamma$ and $L^2$ are conjugate by a multiplication operator, and the conjugate operator to $\\Delta$ is a compact perturbation of the Laplacian, which establishes Fredholm properties. \n\n\n\\subsection{ Kondratiev spaces}\n\nKondratiev spaces were first introduced to study boundary value problems for elliptic equations in domains with conical points \\cite{kondrat1967boundary}. Nirenberg and Walker \\cite{nirenberg1973null} later used these spaces to show that elliptic operators with coefficients that decay sufficiently fast at infinity have finite dimensional kernel when considered as operators between these weighted spaces and McOwen \\cite{McOwen} established Fredholm properties for the Laplacian. Lockhart and McOwen \\cite{lockhart1981fredholm, lockhart1983elliptic, lockhart1985elliptic} built on these ideas to establish Fredholm properties for classes of elliptic operators. For example, Lockart \\cite{lockhart1985elliptic} studied elliptic operators of the form $A = A_{\\infty} +A_0$ in non-compact manifolds, where $A_{\\infty}$ represents a constant coefficient homogeneous elliptic operator of order $m$, and $A_0$ an operator of order at most $m$ with coefficients that decay fast at infinity. More recently, Kondratiev spaces \nwere used to study the Laplace operator in exterior domains \\cite{amrouche2008mixed} and similar weighted spaces were used in \\cite{milisic2013weighted} to understand the Poisson equation in a 1 periodic infinite strip $Z = [0,1] \\times \\mathbb{R}$. \n\nWe will denote Kondratiev spaces by $M^{s,p}_{\\gamma} $ and define them as the completion of $C^{\\infty}_0(\\mathbb{R}^n)$ under the norm\n\\[ \n\\| u\\|_{M^{s,p}_{\\gamma}} = \\sum_{|\\alpha|\\leq s} \\| \\langle {\\bf x} \\rangle^{\\gamma + |\\alpha|} D^{\\alpha} u \\|_{L^p},\n\\]\nwhere $\\langle {\\bf x} \\rangle = ( 1 + |{\\bf x}|^2)^{1\/2}$, $\\gamma \\in \\mathbb{R}$, $ s \\in \\mathbb{N}$, and $p \\in (1, \\infty)$. Notice the embeddings $M^{s,p}_{\\gamma} \\hookrightarrow W^{s,p}_{\\gamma}$, as well as $M^{s,p}_{\\gamma} \\hookrightarrow M^{s-1,p}_{\\gamma}$. \n\nThe following theorem describes the behavior of the Laplacian in Kondratiev spaces. Its proof can be found in \\cite{McOwen}.\n \n\\begin{thm}\\label{McOwen}\nLet $10$. A reasonable Ansatz then is\n\\[ \nA(x,y, \\varepsilon) = (\\sqrt{1-k^2} + s(x,y;\\varepsilon) ) \\mathrm{e}^{\\mathrm{i}(kx + \\phi( x,y;\\varepsilon))},\n\\]\nwith new variables $s,\\phi$, which solve\n\\begin{align}\\label{amp1}\n\\Delta s + (s+\\tau) - ( s + \\tau)( k^2 + 2k \\partial_x \\phi + | \\nabla \\phi|^2) - (s + \\tau)^3 + \\varepsilon \\Real( g \\mathrm{e}^{-\\mathrm{i}(kx + \\phi)})&=0\\\\\n\\Delta \\phi + \\frac{2k \\partial_x s}{s + \\tau} + \\frac{2 \\nabla s \\cdot \\nabla \\phi}{s +\\tau} + \\frac{ \\varepsilon \\Imag(g \\mathrm{e}^{-\\mathrm{i}(kx + \\phi)})}{s + \\tau}&=0,\\label{phase1}\n\\end{align}\nwhere we set $\\tau=\\sqrt{1-k^2}$. Linearizing at $\\varepsilon=0, s=0,\\phi=0$, we obtain the operator\n\\begin{equation}\\label{linearoperator}\n L \\begin{bmatrix} s\\\\ \\phi \\end{bmatrix} = \\begin{bmatrix} \n\\Delta - 2\\tau^2 & -2k\\tau \\partial_x \\\\ \n \\dfrac{2k }{\\tau} \\partial_x & \\Delta \n\\end{bmatrix} \\begin{bmatrix} s\\\\ \\phi \\end{bmatrix}.\n\\end{equation}\n\n\nThe results from the Section \\ref{weightedspaces}, and in particular Theorem \\ref{McOwen}, suggest that we should require that $\\phi\\in M^{2,p}_{\\gamma}$ and that \\eqref{phase1} holds in $L^p_{\\gamma+2}$. \nThen $\\phi_x\\in W^{1,p}_{\\gamma+1}$ and, using the linearization of \\eqref{amp1} with Proposition \\ref{Sobolev}, this suggests $s\\in W^{3,p}_{\\gamma+1}$ and $s_x\\in W^{2,p}_{\\gamma+1}$. This however is not sufficient localization for \\eqref{phase1}, where $s_x$ enters, and which we assumed to be satisfied in $L^p_{\\gamma+2}$. \n\nIn other words, the coupling terms, which are absent for $k=0$, prohibit the simple use of Sobolev spaces for $s$ and Kondratiev spaces for $\\phi$. Roughly speaking, the coupling destroys the linear scaling invariance in the $\\phi$-equation, which is necessary at least at infinity in results on Fredholm properties in Kondratiev spaces, which intrinsically mix regularity and localization properties. We intend to address these problems more generally in future but focus here on a simple an direct construction that circumvents the problem by extending the system and introducing appropriate norms for derivatives. \n\nConsider therefore $ T: \\mathcal{D} \\subset \\mathcal{X} \\rightarrow \\mathcal{R} \\subset \\mathcal{Y}$,\n\\begin{equation}\\label{extendedlinear}\n T \\begin{bmatrix} s \\\\ \\psi \\\\ \\theta \\\\ u \\\\ v\\\\ w \\end{bmatrix} = \n \\begin{bmatrix}\n \\Delta -a & -1 & 0 & 0 & 0 & 0 \\\\\n 0 & \\Delta & 0 & b & 0 & 0\\\\\n 0 & 0 & \\Delta & 0 & b & 0\\\\\n 0 & -\\partial_{xx} & 0 & \\Delta -a & 0 & 0 \\\\\n 0 & 0 & -\\partial_{xx} & 0 & \\Delta -a & 0 \\\\\n 0 & -\\partial_{yy} & 0 & 0 & 0 & \\Delta -a \n \\end{bmatrix}\\begin{bmatrix} s \\\\ \\psi \\\\ \\theta \\\\ u \\\\ v\\\\ w \\end{bmatrix},\n\\end{equation}\nwere \\hskip0.3cm $\\psi = 2k \\tau\\partial_x \\phi$,\\hskip0.3cm $\\theta= 2k \\tau \\partial_y \\phi$,\\hskip0.3cm $a = 2\\tau^2$,\\hskip0.3cm $b = 4k^2$, \n\\[\n\\mathcal{X} = W^{2,2}_{\\gamma} \\times M^{2,p}_{\\gamma} \\times M^{2,p}_{\\gamma} \\times L^p_{\\gamma+2}\\times L^p_{\\gamma+2}\\times L^p_{\\gamma+2},\\] \n\\[ \\mathcal{Y} = L^p_{\\gamma} \\times L^p_{\\gamma+2} \\times L^p_{\\gamma+2} \\times W^{-2,p}_{\\gamma+2}\\times W^{-2,p}_{\\gamma+2}\\times W^{-2,p}_{\\gamma+2}.\n\\]\nHere, $W^{-k,p}_\\gamma$ denotes the dual of $W^{k,p}_\\gamma$. We also define the closed subspaces\n \\[ \\mathcal{D} = \\left \\{ (s, \\psi, \\phi, u,v,w) \\in \\mathcal{X}: u = \\partial_{xx} s, \\quad v= \\partial_{xy} s,\\quad w = \\partial_{yy} s, \\quad \\partial_y \\psi = \\partial_x \\theta \\right \\}, \\]\n\\begin{align*}\n \\mathcal{R} & = \\left \\{ y=(f_1,f_2,f_3,f_4,f_5,f_6) \\in \\mathcal{Y}: \\int f_2 = \\int f_2 \\cdot y= \\int f_3= \\int f_3 \\cdot x=0, \\right. \\\\\n& \\quad \\left.f_4 = \\partial_{xx} f_1, f_5= \\partial_{xy} f_1, f_6 = \\partial_{yy}f_1, \\quad \\text{and} \\quad \\partial_y f_2 = \\partial_x f_3 \\right \\}.\n\\end{align*}\nThe second and third equation in (\\ref{extendedlinear}) are obtained by taking the $x$ and $y$ derivatives of the phase equation. The last three equations come from taking the second order partial derivatives of the amplitude equation with respect to $xx$, $xy$, and $yy$.\n \n We will see in Section \\ref{sectionlinear} that the linear operator $T: \\mathcal{D} \\rightarrow \\mathcal{R}$ is a Fredholm operator of index $-1$ for optimal choices of weights, indicating a missing parameter in the far field. We therefore add a single degree of freedom in the far field through the variable $\\hat{c}\\in\\mathbb{R}$ via the Ansatz \n\\[\n\\begin{array}{l c l c c c l c l\ns & =& \\hat{s} + \\hat{c} P_1, & & & &u &=& \\hat{u} + \\hat{c} \\partial_{xx} P_1,\\\\\n\\psi & =& \\hat{\\psi} + \\hat{c} \\partial_x P_2, & & & &v &=& \\hat{v} + \\hat{c} \\partial_{xy} P_1,\\\\\n\\theta & =& \\hat{\\theta} + \\hat{c}\\partial_y P_2, & & & &w &=& \\hat{w} + \\hat{c} \\partial_{yy} P_1,\n\\end{array}\n\\]\nwhere \n\\begin{equation}\\label{e:p1p2}\nP_1= \\dfrac{1- \\alpha}{2 b \\alpha}\\partial_x[\\chi \\log(\\alpha x^2+y^2)], \\qquad P_2 = \\dfrac{1}{2}\\chi \\log(\\alpha x^2+y^2),\n\\end{equation}\n$b = (2k)^2$, $\\alpha = \\dfrac{1-k^2}{1-3k^2}$, and $\\chi$ is a smoothed version of the indicator function $\\chi_{|\\mathbf{x}|>1}$. Substituting this Ansatz into \\eqref{amp1},\\eqref{phase1} and linearizing, we find an operator $\\hat{T}: \\mathcal{D} \\times \\mathbb{R} \\rightarrow \\mathcal{R}$, given by \n\\begin{equation}\\label{newT} \\hat{T} \\xi =\n \\begin{bmatrix} \n \n \\Delta -a & -1 & & & & &\\Delta P_1 \\\\\n & \\Delta & &b & & &\\Delta P_2 +b \\partial_{xx}P_1 \\\\\n & &\\Delta & & b& & \\Delta P_3 + b \\partial_{xy} P_1 \\\\\n & - \\partial_{xx} & & \\Delta -a & & & \\Delta \\partial_{xx} P_1\\\\\n & &- \\partial_{xx} & & \\Delta -a & & \\Delta \\partial_{xy} P_1 \\\\\n& - \\partial_{yy} & & & &\\Delta -a & \\Delta \\partial_{yy} P_1 \\\\\n \\end{bmatrix} \\begin{bmatrix} \\hat{s} \\\\ \\hat{\\psi} \\\\ \\hat{\\theta} \\\\ \\hat{u} \\\\ \\hat{v} \\\\ \\hat{w} \\\\ \\hat{c} \\end{bmatrix},\n \\end{equation}\nwhere again $a = 2\\tau^2$, and $b = 4k^2$. We will show in the Section \\ref{sectionlinear} that this operator is invertible. \n\nRecapitulating, we are lead to consider the Ansatz\n\\begin{equation}\\label{Ansatz}\nA(x,y; \\varepsilon, \\varphi) = \\left(\\sqrt{1 -k^2} + s(x,y;\\varepsilon, \\varphi)+ c(\\varepsilon, \\varphi) P_1(x,y)\\right) \\mathrm{e}^{ \\mathrm{i} \\left (kx + \\phi(x,y;\\varepsilon, \\varphi)+ \\frac{c(\\varepsilon, \\varphi)}{2k \\sqrt{1-k^2}} P_2(x,y) \\right)}, \n\\end{equation}\nwith $P_1$ and $P_2$ as in \\eqref{e:p1p2}, with additional equations for the derivatives \n\\[ \nu = \\partial_{xx} s, \\quad v = \\partial_{xy} s, \\quad w = \\partial_{yy} s, \\quad \\psi = (2k) \\partial_{x} \\phi,\\quad \\theta = (2k) \\partial_y \\phi.\n\\]\nWe obtain a nonlinear equation \n\\begin{equation}\\label{nl0}\n\\hat{F}_{\\varepsilon, \\varphi}=0, \\qquad \\hat{F}_{\\varepsilon, \\varphi} : \\mathcal{D} \\times \\mathbb{R}^3 \\rightarrow \\mathcal{R};\n\\end{equation}\nsee Subsection \\ref{sectionnonlinear} for a more detailed description of this nonlinear equation. The advantage of this subtle reformulation of the problem is encoded in the following result, which establishes applicability of the standard Implicit Function Theorem and is the key ingredient to the proof of Theorem \\ref{MainTh1}. \n\n\n\n\\begin{thm}\\label{MainTh}\nLet $p=2$, $\\gamma \\in (0,1)$, and $g \\in W^{2,p}_{\\beta}$, with $\\beta > \\gamma+2$. Then, the operator $\\hat{F}_{\\varepsilon,\\varphi}: \\mathcal{D} \\times \\mathbb{R}^3 \\rightarrow \\mathcal{R}$ is of class $C^\\infty$. Furthermore, for fixed $\\varphi$ and at $\\varepsilon =0$, its derivative is given by the invertible operator $\\hat{T}: \\mathcal{D} \\times \\mathbb{R} \\rightarrow \\mathcal{R}$.\n\\end{thm}\n\nThe proof of this theorem will occupy the next 2 Sections. In Section \\ref{sectionlinear} we show the fact that $T$ is a Fredholm operator and that $\\hat{T}$ is invertible, and in Section \\ref{sectionnonlinear} we show that the operator $\\hat{F}$ is of class $C^{\\infty}$.\n\n\n\n\n\n\n \n\\section{The linear operator}\\label{sectionlinear}\nIn this section we consider the linear operators $T: \\mathcal{D} \\rightarrow \\mathcal{R}$ and $\\hat{T}:\\mathcal{D} \\times \\mathbb{R} \\rightarrow \\mathcal{R}$ defined in (\\ref{extendedlinear}) and (\\ref{newT}), respectively. We first prove that for $p=2$ and $\\gamma \\in (0,1)$ the operator $T: \\mathcal{X} \\rightarrow \\mathcal{Y} $ is a Fredholm operator of index $-6$. Then, we show that restricting the domain and range to $\\mathcal{D}$ and $\\mathcal{R}$ turns $T$ into a Fredholm operator of index $-1$. Finally, using a bordering lemma, we prove that the operator $\\hat{T}$ is invertible. Throughout, we assume $ \\gamma \\in (0, 1)$ and $0<|k|< \\dfrac{1}{\\sqrt{3}}$.\n\n\\begin{prop}\\label{FredholmT1} \\\nThe operator $T:\\mathcal{X} \\rightarrow \\mathcal{Y}$ defined in (\\ref{extendedlinear}) is a Fredholm operator with index $i=-6$ and trivial kernel. The cokernel is spanned by \n\\begin{equation}\\label{sol}\n\\{(0,a,0,b,0,0)^Te_j^*, (0,0,a,0,b,0)^Te_j^*, \\, j=1,2,3\\},\\qquad e_1^*=1,e^*_2=x,e^*_3=y.\n\\end{equation}\n\n\\end{prop}\n\n\\begin{proof} \nFirst, notice that due to the lower block-triangular structure of $T$, it is enough to consider the restriction $\\tilde{T}$ to the variables $\\psi, \\theta, u$, $v$, which we write in the form \n\\begin{equation}\\label{Perturbed} \n\\tilde{T}\\xi= L \\xi+b B\\xi, \n\\end{equation}\nwhere $b = 4k^2$, $a = 2(1-k^2)$, \n\\[ \nL= \\begin{bmatrix} \\Delta & 0 &0&0 \\\\ 0& \\Delta&0 &0\\\\ - \\partial_{xx} &0& \\Delta -a&0 \\\\ 0 &-\\partial_{xx} &0 &\\Delta -a \\end{bmatrix}, \\qquad\nB = \\begin{bmatrix} 0 &0 & I& 0 \\\\ 0 & 0 &0 &I\\\\ 0&0&0&0\\\\0&0&0&0 \\end{bmatrix}, \\qquad\n\\text{and} \\quad\\xi = \\begin{bmatrix} \\psi \\\\ \\theta \\\\ u \\\\ v \\end{bmatrix}. \n\\]\nWe need to show that \n\\[\n\\tilde{T}: \\tilde{\\mathcal{X}} = M^{2,2}_{\\gamma} \\times M^{2,2}_{\\gamma} \\times L^2_{\\gamma+2} \\times L^2_{\\gamma+2}\\longrightarrow\\tilde{\\mathcal{Y}} = L^2_{\\gamma+2} \\times L^2_{\\gamma+2} \\times W^{-2,2}_{\\gamma+2} \\times W^{-2,2}_{\\gamma+2},\n\\]\nis a Fredholm operator of index $i =-6$. Since $\\tilde{T}$ is block-diagonal with respect to $(\\psi,u)$ and $(\\theta,v)$, it is sufficient to show that the restriction to $(\\psi,u)$ is Fredholm with index $-3$. For $b=0$, the claim now follows directly from Theorem \\ref{McOwen} and Proposition \\ref{Sobolev}, due to the lower triangular structure and the fact that, with our choice of $\\gamma,p$, the Laplacian is Fredholm with index $-3$. We will address the more difficult situation $b\\neq 0$, next.\n\nIn order to establish the desired Fredholm properties, we need to solve \n\\begin{align}\\label{e1}\n \\Delta \\psi + b u & = f_1\\\\ \\label{e2}\n \\partial_{xx} \\psi + (\\Delta -a) u &= f_2,\n\\end{align}\nfor $f_1,f_2$ in a codimension-3 subspace of $L^2_{\\gamma+2} \\times W^{-2,2}_{\\gamma+2}$, with bounds on $(\\psi,u)\\in M^{2,2}_{\\gamma} \\times L^2_{\\gamma+2}$. \n\n\nDenote by $I-Q$ a projection on the range of the Laplacian, so that $\\int (I-Q) f=\\int x (I-Q) f=\\int y (I-Q) f =0$. We can then decompose \n\\begin{align}\\label{e3}\n \\Delta \\psi + b u & = (I-Q)f_1\\\\ \\label{e4}\n \\partial_{xx} \\psi + (\\Delta -a) u &= (I-Q)f_2,\n\\end{align}\nand $Q u=\\frac{1}{b} Q f_1= - \\frac{1}{a}Q f_2$, exhibiting the 3 solvability conditions \\eqref{sol}. We next solve \\eqref{e4} for $u$, substitute in \\eqref{e3}, to obtain\n\\[\n\\mathcal{L} \\psi=(I-Q)f_1-(\\Delta-a)^{-1}(I-Q)f_2=:f,\\qquad \\mathcal{L}=[\\Delta + b (\\Delta -a)^{-1} \\partial_{xx}],\n\\]\nwhere $f=(I-Q)f$. It therefore remains to show that $\\mathcal{L}:M^{2,2}_{\\gamma}\\to (I-Q)L^2_{\\gamma+2}$ is invertible.\nWe therefore factor \n\\[ \\mathcal{L} \\psi = \\mathcal{M} \\left( \\Delta - \\frac{b}{a} \\partial_{xx}\\right) \\psi, \\]\n\\begin{center}\n\\begin{tikzpicture} \n\\matrix(m)[matrix of math nodes,\n row sep=3.5em, column sep=5.5em,\n text height=2.5ex, text depth=0.25ex]\n {M^{2,2}_{\\gamma}&(I-Q) L^2_{\\gamma+ 2}&(I-Q) L^2_{\\gamma+2} \\\\};\n \\path[->,font=\\footnotesize,>=angle 90] \n (m-1-1) edge node[auto] {$\\Delta - \\dfrac{b}{a} \\partial_{xx}$} (m-1-2)\n\t (m-1-2) edge node[auto] {$\\mathcal{M}$} (m-1-3); \n\\end{tikzpicture}\n\\end{center}\nBy Theorem \\ref{McOwen} the operator $\\left(\\Delta - \\dfrac{b}{a} \\partial_{xx} \\right) : M^{2,2}_{\\gamma} \\rightarrow R_m$ is invertible, since it is conjugate to the Laplacian by a simple $x$-rescaling operator. It is therefore sufficient to establish that $\\mathcal{M}$ is an isomorphism of $(I-Q)L^2_{\\gamma+2}$. \nConsider therefore the associated Fourier symbol \n\\[\n\\hat{\\mathcal{M}}(k,l) = \\frac{k^2+l^2}{k^2+l^2 -\\frac{b}{a} k^2} - \\frac{bk^2}{(k^2+l^2+a) ( k^2+l^2 - \\frac{b}{a}k^2)}.\n\\]\nExploiting that $k^2<\\frac{1}{3}$ so that $1-\\dfrac{b}{a} >0$, it is straightforward to see that \n\\[\n\\sup_{(k,l) \\in \\mathbb{R}^2} |\\hat{\\mathcal{M}}(k,l)|+|\\hat{\\mathcal{M}}(k,l)^{-1}|<\\infty,\n\\]\nso that $\\mathcal{M}$ is an isomorphism of $L^2$. We next show that $\\mathcal{M}$ is an isomorphism on $(I-Q)L^2_j$, $j=2,3$, which by interpolation theory gives the desired result. Equivalently, we need to show boundedness of the multiplication operator $\\hat{\\mathcal{M}}$ on the subspace of $H^j$, $j=2,3$ consisting of functions functions $f$ with $f(0)=0$ and, in case $j=3$, $\\nabla f(0)=0$. Since $\\hat{\\mathcal{M}}(k,l)=a+\\mathrm{O}(k^2+l^2)$ near $k=l=0$, we readily find that $\\| D^{\\alpha} \\hat{\\mathcal{M} } |_{L^{\\infty} }+\\| D^{\\alpha}( \\hat{\\mathcal{M} }^{-1}) |_{L^{\\infty} }< \\infty$ for all indices $|\\alpha|\\leq 2$, which proves that $\\hat\\mathcal{M}$ is an isomorphism on $H^2$. For the case $j=3$, we use that $\\| D^{\\alpha} ( {\\bf k} \\hat{\\mathcal{M} }) |_{L^{\\infty} }+\\| D^{\\alpha}( {\\bf k} \\hat{\\mathcal{M} }^{-1}) |_{L^{\\infty} }< \\infty$ for all indices $|\\alpha|=3$, which readily implies that $\\hat{\\mathcal{M}}$ is an isomorphism on $H^3\\cap\\{f(0)=0\\}$. One can also readily check that the range of $\\mathcal{M}$ and $\\mathcal{M}^{-1}$ is indeed contained in $\\mathrm{Rg}\\,(I-Q)$, which concludes the proof.\n\\end{proof}\n\n\n\\begin{cor}\\label{FredholmT2}\nThe operator $T: \\mathcal{D} \\rightarrow \\mathcal{R}$ defined by (\\ref{extendedlinear}) is Fredholm with index $-1$ and cokernel $\\partial_y f_2+\\partial_x f_3$.\n\\end{cor}\n\n\\begin{proof} \nInspection of $T$ shows that the range of the restriction of $T$ to $\\mathcal{D}$ is actually contained in $\\mathcal{R}$, which implies that $T:\\mathcal{D}\\to\\mathcal{R}$ is injective and the range is closed ($T$ is semi-Fredholm). We need to show that the cokernel is one-dimensional. \n\nTake $f\\in \\mathrm{Rg}\\,(T)\\subset \\mathcal{X}$, with $f_4 = \\partial_{xx} f_1,\\, f_5= \\partial_{xy} f_1, \\, f_6 = \\partial_{yy}f_1$, and $\\partial_y f_2 = \\partial_x f_3$. \nBy construction of $T$, notably having taken derivatives of equations for $s$ and $\\phi$, and by injectivity, $T^{-1}f$ satisfies\n$u = \\partial_{xx} s, \\,v= \\partial_{xy} s$, $w = \\partial_{yy} s$, and $\\partial_y \\psi= \\partial_x \\theta$. As a consequence, the cokernel of $T:\\mathcal{D}\\to\\mathcal{R}$ is a subset of the cokernel of $T:\\mathcal{X}\\to\\mathcal{Y}$. Inspecting the cokernel in \\eqref{sol} and the definition of $\\mathcal{R}$, we see that $(e_j^*,f_4)=(e_j^*,f_5)=0$, so that the integral conditions in the definition of $\\mathcal{R}$ represent precisely 4 of the 6 conditions on the co-kernel. One of the remaining conditions, $\\int f_2\\cdot x=\\int f_3\\cdot y$ is a consequence of $\\partial_y f_2=\\partial_x f_3$, whereas $\\int f_2\\cdot x$ can be readily seen to act non-trivially. As a consequence $T$ is Fredholm of index -1 as claimed, and the cokernel is spanned by $(0,x,0,0,0,0)^T$ or, equivalently, $(0,x,y,0,0,0)^T$.\n\\end{proof}\n\nWe next consider the operator $\\hat{T}: \\mathcal{D} \\times \\mathbb{R} \\rightarrow \\mathcal{R}$ defined by (\\ref{newT}). Recall that\n\\begin{equation}\\label{changevariables}\n\\begin{array}{l c l c c c l c l\ns & =& \\hat{s} + \\hat{c} P_1, & & & &u &=& \\hat{u} + \\hat{c} \\partial_{xx} P_1,\\\\\n\\psi & =& \\hat{\\psi} + \\hat{c} P_2, & & & &v &=& \\hat{v} + \\hat{c} \\partial_{xy} P_1,\\\\\n\\theta & =& \\hat{\\theta} + \\hat{c} P_3, & & & &w &=& \\hat{w} + \\hat{c} \\partial_{yy} P_1.\n\\end{array}\n\\end{equation}\n\n\nHere, \n\\[P_1(x,y) = \\dfrac{ (1-\\alpha)}{2b\\alpha} \\partial_x [\\chi \\ln( \\alpha x^2+y^2)],\\quad\n P_2(x,y) = \\dfrac{1}{2} \\partial_x [ \\chi \\ln( \\alpha x^2+y^2)],\\quad\n P_3(x,y) = \\dfrac{1}{2} \\partial_y [ \\chi \\ln( \\alpha x^2+y^2) ],\n\\]\n\n with $\\alpha = \\dfrac{1-k^2}{1-3k^2}$, \\hskip0.4cm $b= (2k)^2$, \\hskip0.4cm and \\hskip0.4cm $\\chi \\in C^{\\infty}(\\mathbb{R}^2)$ defined by\n \\[\\chi(x,y) = \\left\\{ \\begin{array}{c c l} 0 & \\text{if} & 0 \\leq \\sqrt{\\alpha x^2+y^2} \\leq 1\/2\\\\\n \t\t\t\t\t\t1 & \\text{if} & 1 \\leq \\sqrt{\\alpha x^2+y^2}\n\t\t\t\t\t\t\\end{array} \\right. .\\]\n \n\nTo show $\\hat{T}: \\mathcal{D} \\times \\mathbb{R} \\rightarrow \\mathcal{R}$ is invertible we will need the following lemma.\n\n\n\\begin{lem}\\label{extraM}\n\nThe operator \n\\[\nM: \\mathbb{R} \\rightarrow \\mathcal{R},\\qquad c \\mapsto \\begin{bmatrix} \\Delta P_1&\n\t\t\t\t\t\t\t\\Delta P_2 +b \\partial_{xx} P_1 &\n\t\t\t\t\t\t\t \\Delta P_3 + b \\partial_{xy} P_1 &\n\t\t\t\t\t\t\t \\Delta \\partial_{xx} P_1&\n\t\t\t\t\t\t\t \\Delta \\partial_{xy} P_1&\n\t\t\t\t\t\t\t \\Delta \\partial_{yy} P_1\n\t\t\t\t\t\t\t \t\t\t\t\t\t\t \\end{bmatrix}^T c,\n\\]\nis well-defined and its range satisfies \n\\[ \\iint \\Big [\\Delta P_2 +b \\partial_{xx} P_1 \\Big ] \\cdot x \\mathrm{d} x\\mathrm{d} y= \\iint \\Big [ \\Delta P_3 +b \\partial_{xy}P_1 \\Big ] \\cdot y\\mathrm{d} x\\mathrm{d} y \\neq 0 . \\]\n\\end{lem}\n\n\\begin{proof} First notice that the smooth functions $P_i$, for $i = 1,2,3$, are bounded in compact sets and behave like $\\dfrac{1}{|{\\bf x}|}$ for large values of $|{\\bf x} |$ so that the range of $M$ is indeed a subset of $\\mathcal{Y}$. From the definition, it is not difficult to check that the operator $M$ maps into the desired space $\\mathcal{R}$. We need to show that\n\\[ \\iint \\Big [\\Delta P_2 +b \\partial_{xx} P_1 \\Big ] \\cdot x\\mathrm{d} x\\mathrm{d} y = \\iint \\Big [ \\Delta P_3 +b \\partial_{xy}P_1 \\Big ] \\cdot y\\mathrm{d} x\\mathrm{d} y \\neq 0 . \\]\n\nStraightforward calculations, using the rescaling $X = \\sqrt{\\alpha} x , Y = y$, show that \n\\[ \\Delta P_2+ b \\partial_{xx} P_1 = \\frac{\\sqrt{\\alpha}}{2} \\Delta_{X,Y} \\left [ \\frac{\\partial}{\\partial X} \\left ( \\chi \\cdot \\ln ( X^2+Y^2)\\right) \\right ],\\]\nwhere we write $\\Delta_{X,Y} = \\partial_{XX} + \\partial_{YY}$. Therefore,\n\\begin{align*}\n \\iint [\\Delta P_2+ b \\partial_{xx} P_1 ] \\cdot x \\mathrm{d} x \\mathrm{d} y &= \\iint \\left[ \\frac{\\sqrt{\\alpha}}{2} \\Delta_{X,Y} \\left [ \\frac{\\partial}{\\partial X} \\left ( \\chi \\cdot \\ln ( X^2+Y^2)\\right) \\right ] \\right] \\cdot X dX dY \\\\\n &= \\iint \\left[ \\frac{\\sqrt{\\alpha}}{2} \\Delta_{X,Y} \\left ( \\chi \\cdot \\ln ( X^2+Y^2)\\right) \\right] dX dY \\\\\n &= \\sqrt{\\alpha} \\pi .\n \\end{align*}\n\nSimilarly, using the same rescaling, it can be shown that\n\\[\n \\Delta P_3+ b \\partial_{xy} P_1 = \\frac{1}{2} \\Delta_{X,Y} \\left[ \\frac{\\partial}{\\partial Y} \\left( \\chi \\cdot \\ln (X^2+Y^2) \\right) \\right],\\]\nand consequently\n\\begin{align*}\n\\iint \\left[ \\Delta P_3+ b \\partial_{xy} P_1 \\right] \\cdot y \\mathrm{d} x\\mathrm{d} y &= \\iint \\left[ \\frac{1}{2} \\Delta_{X,Y} \\left[ \\frac{\\partial}{\\partial Y} \\left( \\chi \\cdot \\ln (X^2+Y^2) \\right) \\right] \\right] \\cdot Y \\sqrt{\\alpha} dXdY \\\\\n&= \\sqrt{\\alpha} \\pi.\n\\end{align*}\n\\end{proof}\n\n\\begin{cor}\\label{FredholmT3}\nThe operator $\\hat{T}:\\mathcal{D} \\times \\mathbb{R} \\rightarrow \\mathcal{R}$ is invertible.\n\\end{cor}\n\n\\begin{proof} Notice that $\\hat{T}= [ T M]$, where $T: \\mathcal{D} \\rightarrow \\mathcal{R}$ is the Fredholm operator of index $-1$ described in Corollary \\ref{FredholmT2}, and $M: \\mathbb{R} \\rightarrow \\mathcal{R}$ is defined in Lemma \\ref{extraM}. A bordering lemma implies that $\\hat{T}: \\mathcal{D} \\times \\mathbb{R} \\rightarrow \\mathcal{R}$ is a Fredholm operator of index $0$. Lemma \\ref{extraM} implies that $\\mathrm{Rg}\\,(M)\\not\\subset\\mathrm{Rg}\\,(T)$, so that $\\hat{T}$ is onto, hence invertible.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{The nonlinear map}\\label{sectionnonlinear}\n\nIn this section, we show by a series of lemmas that the nonlinear problem \\eqref{nl0} is well defined and continuously differentiable. We first give explicit expressions for each component of the nonlinearities. We then state and prove several lemmas that will help us show that the nonlinearity is well defined. Finally, we show that the nonlinearities are of class $C^{\\infty}$.\n\n\n The following expressions represent each component of the operator $\\hat{F}_{\\varepsilon, \\varphi}$ announced in \\eqref{nl0}:\n\\begin{align*}\n \\hat{F}_1 (\\xi;\\varepsilon, \\varphi) =& \\Delta s - 2\\tau^2 s - (s + \\tau) \\left( \\frac{\\psi}{\\tau} + \\frac{1}{(2k\\tau)^2} (\\psi^2 + \\theta^2)\\right) - (s^3 + 3s^2 \\tau)+ \\varepsilon \\Real[ g \\mathrm{e}^{-\\mathrm{i} (kx +\\phi(\\varphi))}],\\\\\n\\hat{F}_2 (\\xi;\\varepsilon, \\varphi) =& \\Delta \\psi + \\frac{(2k)^2\\tau u + 2( u \\psi + v \\theta + s_x \\psi_x + s_y \\psi_y) }{s+ \\tau} -\\frac{(2 k)^2 \\tau(s_x)^2 + 2s_x( s_x \\psi +s_y \\theta)}{ (s+\\tau)^2} \\\\\n &+ \\frac{\\varepsilon \\Imag[ \\partial_x( g \\mathrm{e}^{-\\mathrm{i} (kx +\\phi(\\varphi))})]}{s+ \\tau} -\\frac{\\varepsilon s_x \\Imag[g \\mathrm{e}^{-\\mathrm{i}(kx +\\phi(\\varphi))}] }{(s+ \\tau)^2},\n\\\\\n\\hat{F}_3 (\\xi;\\varepsilon, \\varphi) =& \\Delta \\theta + \\frac{(2k)^2 \\tau v + 2( v \\psi + w \\theta + s_x \\theta_x +s_y \\theta_y) }{s + \\tau} - \\frac{(sk)^2 \\tau s_xs_y + 2s_y(s_x \\psi + s_y \\theta)}{(s+\\tau)^2} \\\\\n &+ \\frac{ \\varepsilon \\Imag[ \\partial_y( g \\mathrm{e}^{-\\mathrm{i}(kx +\\phi(\\varphi))})]}{s+\\tau} - \\frac{ \\varepsilon s_y \\Imag[ g \\mathrm{e}^{-\\mathrm{i}(kx + \\phi(\\varphi))}]}{(s+ \\tau)^2},\n\\\\\n\\hat{F}_4 (\\xi;\\varepsilon, \\varphi) =& \\Delta u - 2 \\tau^2 u -u\\left( \\frac{\\psi}{\\tau} + \\frac{1}{(2k\\tau)^2} (\\psi^2 + \\theta^2)\\right) -2 s_x \\left( \\frac{\\psi_x}{\\tau} + \\frac{2}{(2k\\tau)^2} (\\psi_x \\psi + \\psi_y \\theta) \\right) \\\\\n& -(s + \\tau) \\left( \\frac{\\psi_{xx}}{\\tau} + \\frac{2}{(2k\\tau)^2} \\left( \\psi_{xx} \\psi + \\psi_{xy} \\theta + | \\nabla \\psi|^2 \\right) \\right)\\\\\n& - ( 6s(s_x)^2 + 3s^2 u + 6 \\tau (s_x)^2 + 6\\tau s u) + \\varepsilon \\Real[ \\partial_{xx}( g \\mathrm{e}^{-\\mathrm{i}(kx +\\phi(\\varphi))})],\n\\\\\n\\hat{F}_5 (\\xi;\\varepsilon, \\varphi) =& \\Delta v - 2 \\tau^2 v -v\\left( \\frac{\\psi}{\\tau} + \\frac{1}{(2k\\tau)^2} (\\psi^2 + \\theta^2)\\right) - s_x \\left( \\frac{\\theta_x}{\\tau} + \\frac{2}{(2k\\tau)^2} (\\theta_x \\psi + \\theta_y \\theta) \\right) \\\\\n& - s_y \\left( \\frac{\\psi_x}{\\tau} + \\frac{2}{(2k\\tau)^2} (\\psi_x \\psi + \\psi_y \\theta) \\right) -(s + \\tau) \\left( \\frac{\\theta_{xx}}{\\tau} + \\frac{2}{(2k\\tau)^2} \\left( \\theta_{xx} \\psi + \\psi_{yy} \\theta + \\theta_x\\psi_x+\\theta_y\\psi_y \\right) \\right)\\\\\n& - ( 6ss_xs_y + 3s^2 v + 6 \\tau s_x s_y + 6\\tau s v) + \\varepsilon\\Real[ \\partial_{xy}( g \\mathrm{e}^{-\\mathrm{i}(kx +\\phi(\\varphi))})],\n\\\\\n\\hat{F}_6 (\\xi;\\varepsilon, \\varphi) =& \\Delta w - 2 \\tau^2 w -w\\left( \\frac{\\psi}{\\tau} + \\frac{1}{(2k\\tau)^2} (\\psi^2 + \\theta^2)\\right) -2 s_y \\left( \\frac{\\theta_x}{\\tau} + \\frac{2}{(2k\\tau)^2} (\\theta_x \\psi + \\theta_y \\theta) \\right) \\\\\n& -(s + \\tau) \\left( \\frac{\\psi_{yy}}{\\tau} + \\frac{2}{(2k\\tau)^2} \\left( \\theta_{xy} \\psi + \\theta_{xy} \\theta + | \\nabla \\theta|^2 \\right) \\right)\\\\\n& - ( 6s(s_y)^2 + 3s^2 w + 6 \\tau (s_y)^2 + 6\\tau s w) +\\varepsilon \\Real[ \\partial_{yy}( g \\mathrm{e}^{-\\mathrm{i}(kx +\\phi(\\varphi))})].\n\\end{align*}\n\nHere, $\\varphi \\in \\mathbb{R}$, $\\tau = \\sqrt{1 - k^2}$ , $k^2< \\dfrac{1}{3}$, and the variable $\\xi = ( s, \\psi, \\theta, u,v,w)$ is given by the formulas in (\\ref{changevariables}), so that we can actually consider $\\hat{F}$ as an operator on $\\mathcal{D}\\times \\mathbb{R}$ for fixed $\\varepsilon,\\varphi$.\n\n Since $(2k\\tau) \\nabla \\phi = \\langle \\psi, \\theta \\rangle$ we define $\\phi$ by\n\\begin{equation}\\label{defphi} \\phi(x,y; \\varepsilon, \\varphi) = \\phi_\\mathrm{bd}+ \\phi_{\\log}, \\end{equation}\nwhere\n\\begin{align*}\n \\phi_\\mathrm{bd}(x,y;\\varepsilon,\\varphi) = &\\varphi + \\frac{1}{2k\\tau}\\int_{t=0}^1 \\left(\\hat{\\psi}(tx,ty;\\varepsilon)x+ \\hat{\\theta} (tx,ty;\\varepsilon)y\\right) \\\\\n \\phi_{\\log}(x,y;\\varepsilon,\\varphi) =& \\frac{1}{2k\\tau} \\int_{t=0}^1\\left( P_2(tx,ty)x+ P_3 (tx,ty)y\\right) = \\frac{1}{2k\\tau} \\chi \\log(\\alpha x^2+y^2).\n \\end{align*}\n The following lemma shows that $\\phi_\\mathrm{bd}$ is a well-defined function.\n\\begin{lem}\\label{phasebounded}\nIf $\\hat{\\psi}, \\hat{\\theta} \\in M^{2,2}_{\\gamma}$ then for fixed $\\varepsilon$ and $\\varphi$, the function $\\phi_\\mathrm{bd}(x,y; \\varepsilon, \\varphi)$ is well defined, bounded, continuous, and approaches a constant $\\varphi + \\Phi_{\\infty}(\\varepsilon)$ as $| {\\bf x} |\\rightarrow \\infty$.\n\\end{lem}\n\\begin{proof} Note that $\\phi$ is continuous since $\\hat{\\psi}, \\hat{\\theta} \\in M^{2,p}_{\\gamma} \\subset BC^0$.\nLemma \\ref{decay} guarantees that for large $|{\\bf x} |$ we have $|\\hat{\\psi}|, |\\hat{\\theta}| \\leq C | {\\bf x} | ^{-\\gamma-1}$. Therefore, the integrals converge as $| {\\bf x} |\\to \\infty $.\n\\end{proof}\n\nThe next four lemmas will help us show that the operator $\\hat{F}_{\\varepsilon, \\varphi}: \\mathcal{D} \\times \\mathbb{R}^3 \\rightarrow \\mathcal{R}$ is well defined.\n\n\n\\begin{lem}\\label{littleimbedding}\nThere exists $C>0$ so that for all $u \\in L^p_{\\gamma}$ with $ Du \\in L^p_{\\gamma +1}$, $\\langle {\\bf x} \\rangle^{\\gamma}u \\in W^{1,p}$ and, \n\\[\n\\|\\langle {\\bf x} \\rangle^{\\gamma}u\\|_{W^{1,p}}\\leq C \\|u\\|_{L^p_{\\gamma}}+\\|Du\\|_{L^p_{\\gamma +1}}.\n\\]\n\\end{lem}\n \n \\begin{proof} We need to show that $D( \\langle {\\bf x} \\rangle^{\\gamma} u) \\in L^p$. We compute\n \\[ D( \\langle {\\bf x} \\rangle^{\\gamma} u ) = D u \\cdot \\langle {\\bf x} \\rangle^{\\gamma}+ D \\langle {\\bf x} \\rangle^{\\gamma} \\cdot u= Du\\cdot \\langle {\\bf x} \\rangle^{\\gamma} +\\gamma u {\\bf x} (1 + |{\\bf x}|^2) ^{\\frac{\\gamma -2}{2}}.\\]\n Since $Du \\in L^p_{\\gamma+1} \\subset L^p_{\\gamma}$, we conclude that $Du \\cdot \\langle {\\bf x} \\rangle^{\\gamma} \\in L^p$. Furthermore, since $|{\\bf x}|^p \\leq (1 +|{\\bf x}|^2)^{p\/2}$,\n\\[ |u \\cdot D\\langle {\\bf x} \\rangle^{\\gamma} | \\leq |u| |{\\bf x} |\\langle {\\bf x} \\rangle^{(\\gamma-2)} \\leq |u| \\langle {\\bf x} \\rangle^{(\\gamma-1)} \\leq |u| \\langle {\\bf x} \\rangle^{\\gamma } \\in L^p. \\]\nThis implies that $D(\\langle {\\bf x} \\rangle^{\\gamma} u) \\in L^p$ and we obtain $ \\langle {\\bf x} \\rangle^{\\gamma} u\\in W^{1,p}$.\n\\end{proof}\n\n\\begin{lem}\\label{bounded}\nFor $\\gamma >0$, we have the continuous embeddings $M^{2,2}_{\\gamma} \\hookrightarrow W^{2,2}_{\\gamma} \\hookrightarrow W^{2,2}\\hookrightarrow BC^0$.\n\\end{lem}\n\\begin{proof}\nThe first embedding is due to Lemma \\ref{littleimbedding}, the second a consequence of $\\gamma >0$, and the last a classical Sobolev embedding in dimension 2.\n\\end{proof}\n\n\n\\begin{lem}\\label{Holder3}\nFor $\\gamma>0$ there exists $C>0$ such that for all $f,g $ with $\\langle {\\bf x} \\rangle^{\\gamma+1} f, \\langle {\\bf x} \\rangle^{\\gamma+1} g \\in W^{1,p}$,\n\\[\n\\|fg\\|_{L^p_{\\gamma+2}}\\leq C\\|\\langle {\\bf x} \\rangle^{\\gamma+1} f\\|_{W^{1,p}}\\|\\langle {\\bf x} \\rangle^{\\gamma+1} g\\|_{W^{1,p}}.\n\\]\n\\end{lem}\n\n\\begin{proof} By Cauchy-Schwartz,\n\\[\n\\|fg\\langle {\\bf x} \\rangle^{(\\gamma+2)}\\|_{L^p}\\leq \\|f\\langle {\\bf x} \\rangle^{(1+(\\gamma\/2))}\\|_{L^{2p}} \\|g\\langle {\\bf x} \\rangle^{(1+(\\gamma\/2))}\\|_{L^{2p}}\n\\]\nwhich proves the lemma using $\\gamma>0$ and the Sobolev embedding $W^{1,p}\\hookrightarrow L^{2p}$, in $n=2$.\n\\end{proof}\n\n\nFredholm properties of the Laplacian imply in particular the following more basic estimate \\cite{nirenberg1973null}.\n\\begin{lem}\\cite[Theorem 3.1]{nirenberg1973null}\\label{nirenberg}\n If $u \\in L^p_{\\gamma}$ and $\\Delta u \\in L^p_{\\gamma+2}$ then $u \\in M^{2,p}_{\\gamma}$ and there exists a constant $C$ such that\n \\[ \\| u \\|_{M^{2,p}_{\\gamma}} \\leq C \\left( \\| u\\|_{L^p_{\\gamma}} + \\| \\Delta u\\|_{L^p_{\\gamma+2}} \\right) .\\]\n\\end{lem}\n\n\n\\begin{rem}\\label{betters}\nNotice that $u,w\\in L^2_{\\gamma+2}$ so that $ \\Delta s =u+w\\in L^2_{\\gamma+2}$, by the above lemma we have that, within the closed subset $\\mathcal{D}\\subset \\mathcal{X}$, $s \\in M^{2,2}_{\\gamma}$ with uniform bounds in terms of $s,u,w$.\n\\end{rem}\n\n\n\nTo proof Theorem \\ref{MainTh}, the following three lemmas establish that each component of the operator $\\hat{F}_{\\varepsilon, \\varphi} : \\mathcal{D} \\times \\mathbb{R}^3 \\rightarrow \\mathcal{R}$ is well defined. Throughout, we use the standing assumptions $0<\\gamma<1$ and $g \\in W^{2,2}_{\\beta}$, with $\\beta >\\gamma +2$.\n\\begin{lem}\nThe component $\\hat{F}_1 : \\mathcal{D} \\times \\mathbb{R}^3 \\rightarrow L^2_{\\gamma}$ is well defined. \n\\end{lem}\n\n\\begin{proof} We can rewrite\n\\[\\hat{F}_1 (\\xi;\\varepsilon, \\varphi) = \\Delta s - 2\\tau^2 s - \\psi - \\frac{s \\psi}{\\tau} - (s + \\tau) ( \\frac{1}{(2k\\tau)^2} (\\psi^2 + \\theta^2)) - (s^3 + 3s^2 \\tau) + \\varepsilon \\Real [ g\\mathrm{e}^{-\\mathrm{i}( kx + \\phi(\\varphi))}]. \\]\n\nA short calculation shows that $\\Delta s - 2\\tau^2 s - \\psi = ( \\Delta - 2\\tau^2) \\hat{s} -\\hat{\\psi} - c \\Delta P_1$ is the first component of $\\hat{T}$, thus well defined. Consider next the term\n$ s \\psi = ( \\hat{s} + c P_1 ) (\\hat{\\psi} + c P_2 ). $\n\nNotice that $\\hat{s}\\psi$ is in $L^2_{\\gamma}$ since both $ P_2$ and $\\hat{\\psi}$, are bounded by Lemma \\ref{bounded} and $\\hat{s} \\in L^2_{\\gamma}$. Since $\\hat{\\psi} \\in L^2_{\\gamma}$ and since $ P_1$ is bounded, $\\hat{\\psi} P_1\\in L^2_{\\gamma}$ as well. Notice also that the product $ P_1\\cdot P_2$ is bounded in compact sets and behaves like $ \\dfrac{1}{|{\\bf x}|^2} $ for large values of $|{\\bf x}|$, hence it belongs to $L^2_{\\gamma}$ provided $\\gamma<1$. This shows the term $ s \\psi $ is in the desired space.\n\nUsing similar arguments, it is easy to check that the functions $\\psi^2, \\theta^2, s^3,s^2$ and $s( \\psi^2+ \\theta^2)$ are in $L^2_\\gamma$. Finally, by Lemma \\ref{phasebounded} we know $\\phi$ is a bounded continuous function so that we can conclude $\\mathrm{e}^{-\\mathrm{i}(kx + \\phi(\\varphi))}$ is a well defined function in $L^{\\infty}$. This implies that the term $\\Real [ g\\mathrm{e}^{-\\mathrm{i}(kx + \\phi(\\varphi) )}]$ is in $L^2_{\\gamma}$ since $g \\in L^2_{\\beta} \\subset L^2_{\\gamma}$.\n\\end{proof}\n\n\n\n\n\n\\begin{lem}\\label{welldefinedF1}\nThe component $\\hat{F}_2 : \\mathcal{D} \\times \\mathbb{R}^3 \\rightarrow L^2_{\\gamma+2}$ is well defined.\n\\end{lem}\n\n\\begin{proof} Since we are trying to find solutions near $\\xi=0$ we can assume $s\/\\tau$ is close to zero. We can therefore write\n\\begin{align*}\n \\Delta \\psi + (2k)^2 \\frac{\\tau u}{s +\\tau} &= \\Delta \\psi + (2k)^2 \\left(u+\\frac{su}{\\tau+su}\\right)\\\\\n & = \\Delta \\hat{\\psi} + (2k)^2( \\hat{u} + c \\Delta P_2 + (2k)^2 \\partial_{xx} P_2 ) + (2k)^2\\frac{(\\hat{u} + c \\partial_{xx} P_1))(\\hat{s}+cP_1)}{\\tau+(\\hat{u} + c \\partial_{xx} P_1))(\\hat{s}+cP_1)}.\n \\end{align*}\nNotice that the terms $ \\Delta \\hat{\\psi} + (2k)^2 \\hat{u} + c [ \\Delta P_2 + (2k)^2 \\partial_{xx}P_2 ], $ \nrepresent the second component of the linear operator $\\hat{T}: \\mathcal{D} \\times \\mathbb{R} \\rightarrow \\mathcal{R}$, hence are well defined. \nIt is now straightforward to see that the remaining nonlinear terms are contained in $L^2_{\\gamma+2}$. In terms of localization, the most dangerous term is $P_1\\partial_{xx}P_1$, which can be bounded as \n\\[ \\int \\left | P_1 \\partial_{xx} P_1 \\right|^2 \\langle {\\bf x} \\rangle^{2(\\gamma+2)} \\leq \\int \\left| \\frac{1}{r^4} \\right|^2 r^{2(\\gamma+2)} r \\mathrm{d} r < \\infty,\\]\nsince $\\gamma<1$.\n\nWe next treat the remaining nonlinearities. Since $s \\in L^{\\infty}$, we only need to show that the numerators in $\\hat{F}_2$ are in $L^2_{\\gamma+2}$. It is not hard to mimic the above arguments to show that the terms $u \\psi$ and $v \\theta$ are in $L^2_{\\gamma+2}$, so we will treat the term $s_x \\psi_x$ first. Using the formulas in (\\ref{changevariables}) we see that\n\\[ s_x \\psi_x = ( \\hat{s}_x + c \\partial_{x} P_1) ( \\hat{\\psi}_x+ c \\partial_x P_2 ).\\]\n\nBy Lemma \\ref{nirenberg} and Remark \\ref{betters} we know that $\\hat{s}\\in M^{2,2}_{\\gamma}$. Therefore, $\\hat{s}_x,\\hat{\\psi}_x\\in W^{1,2}_{\\gamma+1}$ and we can apply Lemma \\ref{Holder3} to conclude that $\\hat{s}_x \\hat{\\psi}_x\\in L^2_{\\gamma+2}$. The remaining terms are easily seen to be in the correct space.\n\n\nSimilar arguments show that the functions $(s_x)^2, s_y \\psi_y, s_xs_y$ are in the correct space and that, since $ \\psi$ and $\\theta$ are bounded by Lemma \\ref{bounded}, $s_x( s_x \\psi+ s_y \\theta)\\in L^2_{\\gamma+2}$. \n\nFinally, because we are assuming that $g$ is in the space $W^{2,2}_{\\beta}$, with $\\beta > \\gamma+2$, $s_x \\in L^2_{\\gamma+1}$, and because the terms $\\psi$ and $\\mathrm{e}^{-\\mathrm{i}(kx + \\phi(\\varphi))}$ are bounded,\n\\[-\\frac{\\varepsilon s_x \\Imag[g \\mathrm{e}^{-\\mathrm{i}(kx +\\phi(\\varphi))}] }{(s+ \\tau)^2} + \\frac{\\varepsilon \\Imag[ \\partial_x( g \\mathrm{e}^{-\\mathrm{i}(kx +\\phi(\\varphi))})]}{s+ \\tau} \\in L^2_{\\gamma+2}.\\]\n Here, we used the fact that $\\psi = (2 k \\tau) \\phi$ so that $ \\partial_x ( g \\mathrm{e}^{-\\mathrm{i}(kx +\\phi(\\varphi))})= [ g_x -i g( k + \\psi\/(2k\\tau)) ] \\mathrm{e}^{-\\mathrm{i}(kx + \\phi(\\varphi))} $.\n\\end{proof}\n\n\n\\begin{lem}\nThe component $\\hat{F}_3 : \\mathcal{D} \\times \\mathbb{R}^3 \\rightarrow L^2_{\\gamma+2}$ is well defined.\n\\end{lem}\n\n\n\\begin{proof} The proof is almost identical to the proof of Lemma \\ref{welldefinedF1} and is omitted here.\n\\end{proof}\n\nFinally, we show\n\n\\begin{lem}\nThe component $\\hat{F}_4 : \\mathcal{D}\\times\\mathbb{R}^3 \\rightarrow W^{-2,2}_{\\gamma+2}$ is well defined. Moreover, the nonlinear part of $\\hat{F}_4$ actually belongs to $L^2_{\\gamma+2}$.\n\\end{lem}\n\n\\begin{proof} We can rewrite $\\hat{F_4}$ as\n\\[ \\Delta u - 2 \\tau^2 u - \\psi_{xx} - \\frac{ s \\psi_{xx}}{\\tau} -u\\left( \\frac{\\psi}{\\tau} + \\frac{1}{(2k\\tau)^2} (\\psi^2 + \\theta^2)\\right) -2 s_x \\left( \\frac{\\psi_x}{\\tau} + \\frac{2}{(2k\\tau)^2} (\\psi_x \\psi + \\psi_y \\theta) \\right) \\]\n\\[ -(s + \\tau) \\left( \\frac{2}{(2k\\tau)^2} \\left( \\psi_{xx} \\psi + \\psi_{xy} \\theta + | \\nabla \\psi|^2 \\right) \\right) - ( 6s(s_x)^2 + 3s^2 u + 6 \\tau (s_x)^2 + 6\\tau s u) + \\varepsilon \\Real[ \\partial_{xx}( g \\mathrm{e}^{-\\mathrm{i}(kx +\\phi(\\varphi))})].\\]\n\nNotice that\n\\[ \\Delta u - 2 \\tau^2 u - \\psi_{xx} = \\Delta\\hat{u} - 2 \\tau^2 \\hat{u} - \\hat{\\psi}_{xx} + c \\Delta( \\partial_{xx} P_1), \\]\n is just the fourth component of the linear operator $\\hat{T}$, thus well defined. Furthermore, notice that $\\hat{\\psi}_{xx} \\in L^2_{\\gamma+2}$. Since $\\Delta \\partial_{xx} P_1$ behaves like $\\dfrac{1}{|{\\bf x}|^5}$ for large $|{\\bf x}|$ and is bounded in compact sets, these two term now also belong to $L^2_{\\gamma+2}$.\nThe arguments used to show that the remaining nonlinearites are in the space $L^2_{\\gamma+2}$ are the same as the once used in the above lemmas, we will omit the details here.\n\\end{proof}\n\n\n\nHaving shown the result for the operator $\\hat{F}_4$ it is not hard to see that the operators $\\hat{F}_5,\\hat{F}_6 : \\mathcal{D}\\times \\mathbb{R}^3 \\rightarrow W^{-2,2}_{\\gamma+2}$ are well defined.\n\n\\begin{rem}\\label{betteru}\nSince all the nonlinear terms are in $L^2_{\\gamma+2}$, including $\\hat{\\psi}_{xx}$ and $\\Delta \\partial_{xx} P_1$, then $\\Delta\\hat{u} - (2\\tau^2)\\hat{u} \\in L^2_{\\gamma+2}$. This implies that for the solution, $\\hat{u} \\in W^{2,2}_{\\gamma+2}$. The same observation holds for $\\hat{v}$ and $\\hat{w}$.\n\\end{rem}\n\n\nIn the next lemma we show that there exist a neighborhood $\\mathcal{U}$ of $\\xi \\in \\mathcal{D} \\times \\mathbb{R}$ such that the operator $\\hat{F}_{\\varepsilon, \\varphi}: \\mathcal{U} \\times \\mathbb{R}^3 \\rightarrow \\mathcal{R}$ is smooth.\n\n\\begin{lem}\\label{differentiable}\nLet $0< \\gamma<1$ and $g \\in W^{2,2}_{\\beta}$, with $\\beta> \\gamma+2$. Then the operator $\\hat{F}_{\\varepsilon, \\varphi} : \\mathcal{D} \\times\\mathbb{R}^3 \\rightarrow \\mathcal{R}$ is of class $C^\\infty$ in a neighborhood the origin.\n\\end{lem}\n\n\\begin{proof} Most nonlinear terms are defined via superposition (or Nemytskii) operators, via smooth algebraic functions, that are automatically smooth once well defined. We therefore concentrate on the term $g \\mathrm{e}^{-\\mathrm{i}\\phi}$ and its derivatives. Recall that\n\\[ \\phi(x,y; \\varepsilon, \\varphi) = \\phi_\\mathrm{bd}+ \\phi_{\\log}, \\]\nwhere\n\\begin{align*}\n \\phi_\\mathrm{bd}(x,y;\\varepsilon,\\varphi) = &\\varphi + \\frac{1}{2k\\tau}\\int_{t=0}^1 \\left(\\hat{\\psi}(tx,ty;\\varepsilon)x+ \\hat{\\theta} (tx,ty;\\varepsilon)y\\right) \\\\\n \\phi_{\\log}(x,y;\\varepsilon,\\varphi) =& \\frac{1}{2k\\tau} \\int_{t=0}^1\\left( P_2(tx,ty)x+ P_3 (tx,ty)y\\right) = \\frac{1}{2k\\tau} c \\chi \\log(\\alpha x^2+y^2).\n\\end{align*}\nIn order to show smoothness, we factor\n\\[\ng \\mathrm{e}^{-\\mathrm{i}\\Phi}=\\left(g\\langle\\mathbf{x}^{\\gamma+2-\\beta}\\rangle\\right)\\cdot\\mathrm{e}^{-\\mathrm{i}\\Phi_\\mathrm{bd}} \\cdot \\left(\\langle\\mathbf{x}^{\\beta-\\gamma-2}\\rangle \\mathrm{e}^{-\\mathrm{i}\\Phi_\\mathrm{log}}\\right)=:G_1\\cdot G_2 \\cdot G_3.\n\\]\nClearly, $G_1\\in L^2_{\\gamma+2}$. By Lemma \\ref{phasebounded}, $\\int\\psi,\\int\\theta\\in L^\\infty$, so that $G_2\\in L^\\infty$ is bounded as a superposition operator. It remains to show that $G_3$ is differentiable with values in $L^\\infty$. This can be readily established, showing that the derivative with respect to $c$ is \n\\[\n\\partial_cG_3=\\langle\\mathbf{x}^{\\beta-\\gamma-2}\\rangle \\mathrm{e}^{-\\mathrm{i}\\Phi_\\mathrm{log}}\\chi \\log(\\alpha x^2+y^2),\n\\]\nhence bounded in $L^\\infty$. Higher derivatives are bounded for the same reasons, which establishes the claim.\n\\end{proof}\n\n\n \n\n\n\\section{Expansions and proof of main result}\\label{Mainresult}\n\nIn this last subsection we use Theorem \\ref{MainTh} to proof Theorem \\ref{MainTh1} and derive the expansions for the stationary solutions to the perturbed Ginzburg-Landau equation near roll patterns. \n\\begin{proof}[of Theorem \\ref{MainTh1}] Recall the Ansatz\n\\[ A(x,y; \\varepsilon, \\varphi) = (\\sqrt{1 -k^2} + s(x,y;\\varepsilon, \\varphi)+ c(\\varepsilon, \\varphi) P_1(x,y)) \\mathrm{e}^{\\mathrm{i} kx + \\mathrm{i} \\phi(x,y;\\varepsilon, \\varphi)+ \\mathrm{i} \\frac{c(\\varepsilon, \\varphi)}{2k \\sqrt{1-k^2}}P_2(x,y)} \\]\nwhere $\\Phi$ was defined in \\eqref{defphi} and $P_1,P_2$ in \\eqref{e:p1p2}.\nFrom Theorem \\ref{MainTh} we know there exists a neighborhood, $\\mathcal{U}$, of $\\mathcal{D} \\times \\mathbb{R}$ where the operator $\\hat{F}_{\\varepsilon,\\varphi}$ is continuously differentiable with invertible derivative at the origin, $\\varepsilon=0$. The Implicit Function Theorem therefore guarantees the existence of solutions $\\xi(\\varepsilon, \\varphi) $ near $\\xi(0, \\varphi)=0$. In particular, we know that $s \\in W^{2,2}_{\\gamma}$, and $\\psi, \\theta \\in M^{2,2}_{\\gamma}$.\n\n\nWe define \n\\[ S(x,y; \\varepsilon, \\varphi) = \\left(\\sqrt{1-k^2} + s(x,y;\\varepsilon, \\varphi) + c(\\varepsilon, \\varphi) P_1(x,y) \\right)\\]\nand\n\\[ \\Phi( x,y;\\varepsilon, \\varphi) =kx + \\phi. \\]\n\nSince $s(x,y;\\varepsilon, \\varphi) \\in W^{2,2}_{\\gamma}$, Lemma \\ref{bounded} ensures that if $s(x,y; \\varepsilon, \\varphi) \\sim \\mathrm{O}(\\langle {\\bf x} \\rangle^{-\\gamma})$. Also, by definition, $P_1(x,y) \\sim \\mathrm{O}(\\langle {\\bf x} \\rangle^{-1})$, and\n\\[ \\lim_{\\mathrm{x}\\to\\infty} S(x,y;\\varepsilon, \\varphi) =S_{\\infty}= \\sqrt{1-k^2}.\\]\n\nBy Lemma \\ref{phasebounded}, $\\phi_\\mathrm{bd} \\to\\varphi + \\Phi_{\\infty}(\\epsilon)$ for $\\mathbf{x}\\to\\infty$ so that \n\\[ \\Phi(x,y;\\varepsilon,\\varphi)-kx-\\frac{c(\\varepsilon,\\varphi)}{2k \\sqrt{1-k^2}}\\log(\\alpha x^2+y^2)\\to \\Phi_\\infty(\\varepsilon)+\\varphi,\\]\nas $|{\\bf x}| \\to \\infty$.\n\n\nTo find an expression for $c(\\varepsilon, \\varphi)$ we expand $\\xi = \\varepsilon \\hat{\\xi} + \\mathrm{o}(\\varepsilon)$. Gathering terms of order $\\varepsilon$ results in the system $\\hat{T} \\hat{\\xi} = \\hat{f}$. Inspecting the second component of this system, we find\n\\[ \\Delta \\hat{\\psi} + b \\partial_{xx} \\hat{u} + \\hat{c} \\left[\\Delta P_2 + b \\partial_{xx} P_1 \\right] = \\dfrac{1}{\\sqrt{1-k^2}}\\Imag[\t ( g_x-ikg) \\mathrm{e}^{-\\mathrm{i}(kx+\\varphi)}].\\]\nTaking the scalar product with $x$ and solving for $\\hat{c}$, we obtain after integration by parts in $x$,\n\\[ \\hat{c} =\\frac{\\sqrt{1-3k^2}}{\\pi(1-k^2)} \\iint \\Imag[ g\\mathrm{e}^{-\\mathrm{i}( kx+ \\varphi)}].\n\\]\nHence $c( \\varepsilon, \\varphi) = \\varepsilon c_1( \\varphi) + \\mathrm{o}(\\varepsilon)$ with $c_1(\\varphi) = \\hat{c}$.\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Appendix}\n\n\\begin{lem}\\label{decay}\nIf $f \\in M^{2,2}_{\\gamma}$ then $|f| \\leq {C} \\| f\\|_{M^{2,2}_{\\gamma}} \\langle {\\bf x} \\rangle^{-\\gamma-1}$ as $| {\\bf x} |\\rightarrow \\infty$.\n\\end{lem}\n\\begin{proof} Since $M^{2,2}_{\\gamma}$ is the completion of $C^{\\infty}_0$ under the norm $\\| \\cdot\\|_{M^{2,2}_{\\gamma}}$, it suffices to show that the result holds for $f \\in C^{\\infty}_0$. In polar coordinates, we have, up to constants \n \\begin{align*}\n \\int |f(\\theta, R)|^2 \\mathrm{d}\\theta &\\leq \\int \\left( \\int_{\\infty}^R |f_r(\\theta, s)| \\mathrm{d} s \\right)^2 \\mathrm{d}\\theta\n =\\int \\left( \\int_{\\infty}^R s^{-\\gamma -3\/2} s^{\\gamma+1} | f_r(\\theta, s)| s^{1\/2} \\mathrm{d} s \\right)^2 \\mathrm{d}\\theta\\\\\n & \\leq \\int \\left(\\int_{\\infty}^R s^{-2(\\gamma + 3\/2)} \\mathrm{d} s \\right) \\left( \\int_{\\infty}^R s^{2(\\gamma+1)} |f_r(\\theta,s)|^2 s\\mathrm{d} s \\right) \\mathrm{d}\\theta \\\\\n &\\lesssim R^{-2(\\gamma+3\/2)+1} \\int \\int_{\\infty}^R s^{2( \\gamma+1)}|f_r(\\theta,s)|^2 s\\mathrm{d} s\\mathrm{d}\\theta ,\n \\end{align*}\nwhich gives\n\\begin{equation}\\label{es1} \\| f(\\cdot, R)\\|_{L^2} \\lesssim R^{-\\gamma-1}\\|\n \\nabla f\\|_{L^2_{\\gamma+1}}.\n\\end{equation}\nSimilarly,\n\\begin{align*}\n\\int |f_{\\theta}(\\theta, R) |^2\\mathrm{d}\\theta &\\leq \\int \\left( \\int_{\\infty}^R |f_{r \\theta}(\\theta,s)|\\mathrm{d} s \\right)^2 \\mathrm{d}\\theta\n= \\int \\left( \\int_{\\infty}^R s^{-\\gamma -5\/2} s^{\\gamma+2} |f_{r \\theta}(\\theta,s)| s^{1\/2}\\mathrm{d} s \\right)^2 \\mathrm{d}\\theta\\\\\n& \\leq \\int \\left( \\int_{\\infty}^R s^{-2(\\gamma+5\/2)} \\mathrm{d} s \\right) \\left( \\int_{\\infty}^R s^{2(\\gamma+2)} |f_{r \\theta}(\\theta,s)|^2 s\\mathrm{d} s \\right) \\mathrm{d}\\theta \\\\\n& \\lesssim R^{-2(\\gamma+5\/2)+1} \\int \\int_{\\infty}^R s^{2(\\gamma+2)} |f_{r \\theta}(\\theta,s)|^2 s\\mathrm{d} s \\mathrm{d}\\theta.\n\\end{align*}\nThis gives\n\\begin{equation}\\label{es2} \\| f_\\theta(\\cdot, R)\\|_{L^2}\\leq R^{-\\gamma-2} \\| f_{r\\theta} \\|_{L^2_{\\gamma+2}}.\\end{equation}\nCombining \\eqref{es1} and \\eqref{es2} and using the interpolation inequality \\cite[Thm 5.9]{adams}\n\\[\n\\|f(\\cdot,R)\\|^2_\\infty\\leq \\|f(\\cdot,R)\\|_{L^2} \\|f(\\cdot,R)\\|_{H^1},\n\\]\nnow proves the claim.\n\\end{proof}\n\n\n\n\n\n\n\n\n \n \n \\bibliographystyle{siam}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \n\nLet $\\DD$ be the open unit disk in the complex plane. Let $L^2$ denote the Lebesgue space of square integrable functions on the unit circle $\\partial\\DD$. The Hardy space $H^2$ is the subspace of $L^2$ of analytic functions on $\\DD$. Let $P$ be the orthogonal projection from $L^2$ to $H^2$.\nFor $\\varphi\\in L^\\infty$, the space of bounded Lebesgue measurable functions on $\\partial\\DD$, the Toeplitz operator $T_{\\varphi}$ and the Hankel operator $H_\\varphi$ with symbol $\\varphi$ are defined on $H^2$ by $$T_{\\varphi}h=P(\\varphi h),$$ and\n\\beq\\label{Han}\nH_{\\varphi}h=U(I-P)(\\varphi h),\n\\eeq\nfor $h\\in H^2$.\nHere $U$ is the unitary operator on $L^2$ defined by $$Uh(z)=\\bz h (\\bz).$$\n\nRecall that the spectrum of a linear operator $T$, denoted as $sp(T)$, is the set of complex numbers $\\lambda$ such that $T - \\lambda I$ is not invertible; here $I$ denotes the identity operator. Let $[T^*,T]$ denote the operator $T^*T-TT^*$, called the self-commutator of $T$. An operator $T$ is called hyponormal if $[T^*,T]$ is positive.\nHyponormal operators satisfy the celebrated Putnam inequality \\cite{put70}\n\\begin{thm}\\label{P}\nIf $T$ is a hyponormal operator, then\n$$\n\\|[T^*,T]\\|\\leq \\frac{Area(sp(T))}{\\pi}.\n$$\n\\end{thm}\n\nNotice that a Toeplitz operator with analytic symbol $f$ is hyponormal, and it is well known that $sp(T_f)=\\overline{f(\\DD)}$. The lower bounds of the area of $sp(T_f)$ were obtained in \\cite{kh(T)} (see \\cite{ax85},\\cite{ale72} \\cite{stan86} and \\cite{stan89} for generalizations to uniform algebras and further discussions). Together with Putnam's inequality such lower bounds were used to prove the isoperimetric inequality (see \\cite{ben14},\\cite{ben06} and the references there). Recently, there has been revived interest in the topic in the context of analytic Topelitz operators on the Bergman space (cf. \\cite{bell14}, \\cite{ol14} and \\cite{fl15}). Together with Putnam's inequality, the latter lower bounds have provided an alternative proof of the celebrated St. Venant's inequality for torsional rigidity.\n\nIn the general case, Harold Widom \\cite{wid64} proved the following theorem for arbitrary symbols.\n\\begin{thm}\nEvery Toeplitz operator has a connected spectrum.\n\\end{thm}\nThe main purpose of this note is to show that for a rather large class of Topelitz operators on $H^2$, hyponormal operators with a harmonic symbol, there is\nstill a lower bound for the area of the spectrum, similar to the lower bound obtained in \\cite{kh(T)} in the context of uniform algebras.\n\nWe shall use the following characterization of the hyponormal Toeplitz operators given by Cowen in \\cite{cow88}\n\\begin{thm}\\label{cow}\nLet $\\varphi\\in L^\\infty$, where $\\varphi=f+\\bar{g}$ for $f$ and $g$ in $H^2$. Then $T_{\\varphi}$ is hyponormal if and only if\n$$\ng=c+T_{\\bh}f,\n$$\nfor some constant $c$ and $h\\in H^\\infty$ with $\\|h\\|_{\\infty}\\leq 1.$\n\\end{thm}\n\n\n\\section{Main Results}\nIn this section, we obtain the lower bound for the area of the spectrum for hyponormal Toeplitz operators by estimating the self-commutators.\n\n\\begin{thm}\\label{2.1}\nSuppose $\\varphi \\in L^\\infty$ and $$\\varphi=f+ \\overline{T_{\\bar{h}}f} ,$$ for $f,h\\in H^\\infty$, $\\|h\\|_{\\infty}\\leq 1$ and $h(0)=0$. Then\n$$\n\\|[T^{*}_{\\varphi}, T_{\\varphi}]\\|\\geq \\int |f-f(0)|^2\\frac{d\\theta}{2\\pi}=||P(\\varphi)-\\varphi(0)||^2_2.\n$$\n\\end{thm}\n\\begin{proof}\nLet\n\\beq\\label{g} g=T_{\\bar{h}}f.\\eeq\nFor every $p$ in $H^2$,\n\\begin{align*}\n\\langle[T^{*}_{\\varphi}, T_{\\varphi}]p,p\\rangle&=\\langle T_{\\varphi}p, T_{\\varphi}p\\rangle-\\langle T^*_{\\varphi}p, T^*_{\\varphi}p\\rangle\\\\\n&=\\langle fp+P(\\bg p), fp+P(\\bg p)\\rangle-\\langle gp+P(\\barf p), gp+P(\\barf p)\\rangle\\\\\n&=||fp||^2-||P(\\barf p)||^2-||gp||^2+||P(\\bg p)||^2\\\\\n&=||\\barf p||^2-||P(\\barf p)||^2-||\\bg p||^2+||P(\\bg p)||^2\\\\\n&=||H_{\\barf} p||^2-||H_{\\bg}p||^2,\n\\end{align*}\nwhere $||\\cdot||$ means the $||\\cdot||_{L^2(\\partial \\DD)}$.\nThe third equality holds because\n$$\n\\langle fp, P(\\bg p)\\rangle=\\langle fp, \\bg p\\rangle=\\langle gp, \\barf p\\rangle=\\langle gp, P(\\barf p)\\rangle.\n$$\nBy the computation in \\cite{cow88}*{p. 4}, \\eqref{g} implies $$H_{\\bg}=T_{\\bk}H_{\\barf},$$\nwhere $k(z)=\\overline{h(\\bz)}$. Thus\n\\beq\\label{rmk}\n\\langle[T^{*}_{\\varphi}, T_{\\varphi}]p,p\\rangle=||H_{\\barf} p||^2-||T_{\\bk}H_{\\barf}p||^2,\n\\eeq\nfor $k\\in H^\\infty$, $\\|k\\|_{\\infty}\\leq 1$ and $k(0)=0$.\n\nFirst, we assume $k$ is a Blaschke product vanishing at $0$. Then\n$$|k|=1\\,\\,\\m{on}\\,\\, \\partial\\DD.$$\nLet $u=H_{\\barf} p\\in H^2$. By \\eqref{rmk} we have\n\\begin{align}\\label{T}\n\\langle[T^{*}_{\\varphi}, T_{\\varphi}]p,p\\rangle&=||u||^2-||T_{\\bk} u||^2=||u||^2-||\\bk u||^2+||H_{\\bk}u||^2=||H_{\\bk}u||^2.\n\\end{align}\nThen\n\\begin{align*}\n||H_{\\bk}u||&=||(I-P)(\\bk u)||=||k\\bu-\\overline{P(\\bk u)}||\\\\\n&\\geq \\sup_{\\substack{m\\in H^2\\\\m(0)=0}}\\frac{|\\langle k\\bu-\\overline{P(\\bk u)},m\\rangle|}{||m||}\\\\\n&={\\sup_{\\substack{m\\in H^2\\\\m(0)=0}}{1 \\over ||m||}\\left|\\int k\\bu \\bm \\,{d\\theta\\over 2\\pi}\\right|}.\n\\end{align*}\nThe last equality holds because $m(0)=0$ implies that $\\bm$ is orthogonal to $H^2$.\nSince $k(0)=0$, taking $m=k$, we find\n\\beq\\label{u}\n||H_{\\bk}u||\\geq \\left|\\int\\bu \\,{d\\theta\\over 2\\pi}\\right|=|u(0)|.\n\\eeq\n\nNext, suppose $k$ is a convex linear combination of Blaschke products vanishing at 0, i.e.\n$$\nk=\\Ga_1B_1+\\Ga_2B_2+...+\\Ga_lB_l,\n$$\nwhere $B_j$'s are Blaschke products with $B_j(0)=0$, $\\Ga_j\\in[0,1]$ and $\\displaystyle\\sum_{j=1}^l \\Ga_j=1$.\n\nBy \\eqref{T} and \\eqref{u}, for each $j$\n\\begin{align*}\n&||u||^2-||T_{\\bar{B_j}}u||^2=||H_{\\bar{B_j}}u||^2\\geq |u(0)|^2\\\\\n\\Longrightarrow\\, &|| T_{\\bar{B_j}}u||\\leq \\sqrt{||u||^2-|u(0)|^2}=||u-u(0)||.\n\\end{align*}\nThen\n\\begin{align}\n\\label{M}||u||^2-||T_{\\bk} u||^2&=||u||^2-||\\Ga_1T_{\\bar{B_1}}u+\\Ga_2T_{\\bar{B_2}}u+...+\\Ga_lT_{\\bar{B_l}}u||^2\\\\\n\\nnb&\\geq ||u||^2-\\Big(\\Ga_1||T_{\\bar{B_1}}u||+\\Ga_2||T_{\\bar{B_2}}u||+...+\\Ga_l||T_{\\bar{B_l}}u||\\Big)^2\\\\\n\\nnb&\\geq ||u||^2- ||u-u(0)||^2=|u(0)|^2.\n\\end{align}\n\nIn general, for $k$ in the closed unit ball of $H^\\infty$, vanishing at $0$, by Carath\\'eodory's Theorem(cf. \\cite{gar81}*{p. 6}), there exists a sequence $\\{B_n\\}$ of finite Blaschke products such that\n$$B_n\\longrightarrow k\\q \\m{pointwise on}\\,\\,\\DD.$$\nSince $B_n$'s are bounded by $1$ in $H^2$, passing to a subsequence we may assume\n$$B_n\\longrightarrow k\\q \\m{weakly in}\\,\\,H^2.$$\nThen by \\cite{rud91}*{Theorem 3.13}, there is a sequence $\\{k_n\\}$ of convex linear combinations of Blaschke products such that\n$$k_n\\longrightarrow k\\q \\m{in}\\,\\,H^2.$$\nSince $k(0)=0$, we can let those $k_n$'s be convex linear combinations of Blaschke products vanishing at $0$.\n\nThen\n$$\n||T_{\\bk_n}u-T_{\\bk} u||=||P(\\bk_n u-\\bk u)||\\leq||k_n-k||\\cdot||u||\\to 0.\n$$\nSince \\eqref{M} holds for every $k_n$, we have\n\\begin{align*}\n\\langle[T^{*}_{\\varphi}, T_{\\varphi}]p,p\\rangle&=||u||^2-||T_{\\bk}u||^2=\\lim_{n\\to\\infty}(||u||^2-||T_{\\bk_n}u||^2)\\\\\n&\\geq|u(0)|^2=|(H_{\\barf} p)(0)|^2.\n\\end{align*}\n\nBy the definition of Hankel operator \\eqref{Han},\n\\begin{align*}\n|(H_{\\barf} p)(0)|&=|\\langle p\\barf, \\bz \\rangle|=\\left|\\int \\barf zp\\,{d\\theta\\over 2\\pi}\\right|.\n\\end{align*}\nFrom the standard duality argument (cf. \\cite{gar81}*{Chapter IV}), we have\n\\begin{align*}\n\\sup_{\\substack{||p||=1 \\\\ p\\in H^2}}\\left|\\int \\barf zp\\,{d\\theta\\over 2\\pi}\\right|&=\\sup\\bigg\\{ \\left|\\int \\barf p\\,{d\\theta\\over 2\\pi}\\right| : p\\in H^2, ||p||=1, p(0)=0 \\bigg\\}\\\\\n&=\\m{dist}(\\barf, H^2)=||f-f(0)||.\n\\end{align*}\nHence\n$$\n||[T^{*}_{\\varphi}, T_{\\varphi}]||=\\sup_{\\substack{||p||=1 \\\\ p\\in H^2}}|\\langle[T^{*}_{\\varphi}, T_{\\varphi}]p,p\\rangle|\\geq ||f-f(0)||^2.\n$$\n\\end{proof}\n\\begin{rem}\nFor arbitrary $h$ in the closed unit ball of $H^\\infty$, it follows directly from \\eqref{rmk} that $T_{\\varphi}$ is normal if and only if $h$ is a unimodular constant. So we made the assumption that $h(0)=0$ to avoid these trivial cases. Of course, Theorem \\ref{2.1} implies right away that $T_\\varphi$ is normal if and only if $f=f(0)$, i.e., when $\\varphi$ is a constant, but under more restrictive hypothesis that $h(0)=0$.\n\\end{rem}\n\n\nApplying Theorem \\ref{P} and \\ref{cow}, we have\n\\begin{cor}\nSuppose $\\varphi \\in L^\\infty$ and $$\\varphi=f+ \\overline{T_{\\bar{h}}f} ,$$ for $f,h\\in H^\\infty$, $\\|h\\|_{\\infty}\\leq 1$ and $h(0)=0$. Then\n$$\nArea(sp(T_\\varphi))\\geq\\pi||P(\\varphi)-\\varphi(0)||^2_2.\n$$\n\\end{cor}\n\n\\begin{rem}\nThus, the lower bound for the spectral area of a general hyponormal Toeplitz operator $T_\\varphi$ on $\\partial \\DD$ still reduces to the $H^2$ norm of the analytic part of $\\varphi$. For analytic symbols this is encoded in \\cite{kh(T)}*{Theorem 2} in the context of Banach algebras. In other words, allowing more general symbols does not reduce the area of the spectrum.\n\\end{rem}\n\n\\section*{Acknowledgments}\nThe first author gratefully acknowledges the hospitality of the Department of Mathematics and Statistics\nat the University of South Florida during the work on the paper.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Indroduction}\n\nGalaxies of similar morphological type exhibit similarities in their \nstructural properties, independent of whether they are isolated in \ngeneral fields or agglomerated in galaxy clusters. These similarities \nunder different environments may emerge due to some {\\it internal} \nregularities that have superseeded external, probabilistic disturbances \nsuch as interactions and\/or mergers.\n\nThere is a well-known evidence from optical observations that many \nedge-on spiral and lenticular galaxies have a universal luminosity profile \nperpendicular to their disk plane, or a universal $z$-distribution of \nstellar mass density if a constant mass-to-luminosity ratio is assumed \n(Tsikoudi 1977; Burstein 1979; van der Kruit \\& Searle 1981). It was \nonce claimed that the $z$-distribution is best fitted by a self-gravitating \nisothermal model like sech$^2(z\/2h_z)$ using a scale parameter $h_z$ \n(van der Kruit \\& Searle 1981). \n\nNear-infrared observations of edge-on spiral galaxies however uncovered an \nexcess over the isothermal model at small $|z|$ where the optical photometry \nis hindered by dust absorption (Wainscoat, Freeman, \\& Hyland 1989; \nAoki et al. 1991; van Dokkum et al. 1994). These infrared observations \nshowed that an exponential distribution $\\exp(-|z|\/h_z)$ provides a superior \nfit to the observed $z$-profiles. Furthermore it is known from analyses of \nstar counts that the vertical structure of the Galactic disk in the solar\nneighbourhood is well fitted by an exponential distribution rather than an \nisothermal model (Bahcall \\& Soneira 1980; Prichet 1983). \n\nAn exponential $z$-distribution can be constructed by adding up several\nstellar disk components with different vertical velocity dispersions, but \nthis is only possible if the contribution from stars with larger velocity \ndispersion is fine-tuned to dominate progressively at larger distances from \nthe disk plane. A mechanism that enables this tuning has not been found \nto date.\n\nThe vertical disk structure has been\nconsidered as a result of dynamical evolution of the stellar component through\nencounters with massive clouds and spiral structures\n(Villumsen 1983; Lacey 1984; Carlberg 1987). However, although collisions with clouds\nmight be important in increasing the velocity dispersions of young stars \nthis effect cannot account for the high velocity dispersion of old stars \n(Lacey 1984). In addition, the resulting disks are isothermal and\nnot exponential.\n\nWe report that the hydrodynamical equations associated with a simple\nstar formation law possess a remarkable solution which naturally produces\nan exponential stellar $z$-distribution in the final stage of gravitational\nsettling of galactic protodisks. This result might provide new insight\ninto the formation of galactic disks and their early evolutionary phases.\n\n\\section{The numerical model}\n\nThe key factor which makes galaxy simulations distinct from either \nhydrodynamical computations is star formation. From theoretical \narguments (Hoyle 1953; Low \\& Lynden-Bell 1976; Silk 1977), the cooling \nof the gas plays a decisive role in the process of star formation as it \nprovides the required cool environments where density fluctuations in the \ngas can become gravitationally unstable on stellar mass scales. It is \ntherefore reasonable to assume that stars form at a rate comparable to the \nlocal cooling rate. It is this working hypothesis that we use in this paper.\n\nWe will restrict ourselves to the gravitational settling of the protodisk\nin the vertical direction, neglecting radial motions. This approximation\nis reasonable because of the observational fact that galactic disks are in\ngeneral rotationally supported. The hydrodynamical equations which describe \nthe evolution of the gas density $\\rho$, the vertical gas velocity $w$ and \nthe thermal gas energy by unit mass $e$ are then\n\\begin{eqnarray}\n \\frac{\\partial \\rho}{\\partial t} + \\frac{\\partial}{\\partial z}\n (\\rho w) & = & - \\frac{\\rho}{t_*} \\nonumber \\\\\n \\frac{\\partial w}{\\partial t} + w \\frac{\\partial w}{\\partial z} +\n \\frac{2}{3} \\frac{\\partial}{\\rho \\ \\partial z} (\\rho e) - g_z & = & 0 \\\\\n \\frac{\\partial e}{\\partial t} + w \\frac{\\partial e}{\\partial z} +\n \\frac{2}{3} e \\frac{\\partial w}{\\partial z} & = & - \\frac{e}{t_c}\n \\;\\;\\;, \\nonumber\n\\end{eqnarray}\nwith\n\\begin{equation}\n\\partial g_z\/\\partial z=-4\\pi G(\\rho+\\rho_*) \\;\\;\\; ,\n\\end{equation}\nwhere $\\rho_*$ is the stellar density and the other notations have their usual\nmeanings. The source terms on the right hand side of equation (1) describe \nthe condensation of gas into stars on a timescale $t_*$ and the gaseous energy \ndissipation on a timescale $t_c=\\rho e\/\\Lambda$ with the cooling rate per unit \nvolume $\\Lambda (\\rho ,e)$. We introduce a free parameter $k=t_*\/t_c$ which \ndetermines the timescale of star formation $t_*$ relative to $t_c$. \nNote that $k$ should be of order unity according to our assumption that star \nformation is related primarily to the cooling activity of the gas.\n\nThe cooling rate is a function of the local gas density and temperature and\ntherefore can be approximated as $\\Lambda = \\Lambda_0 \\rho^a e^b$. Typical \nvalues are $a=2$ and, in an ionized hydrogen gas, $b=+0.5$ or $b=-0.5$ for \nfree-free cooling or free-bound cooling, respectively. Note that possible \nheating would compete with the gas cooling and affect $a$ and $b$. The \npower-law model also includes the possibility that energy dissipation in the \ndisk is dominated by cloud-cloud collisions. In such a case $a=2$ and $b=1.5$ \n(Larson 1969). In order to keep the simulation simple we treat $a$ and $b$ \nas free parameters and do not specify cooling and heating sources in an \nexplicit way.\n\nThe stars will stay most of their time at maximum $z$-distance, that is\napproximately at the position where they formed initially. This is at least\nvalid for a majority of stars that make up the disk where the gas is quickly\nconverted into stars and subsequent contraction of the gas no longer has a \nsignificant dynamical effect on the stellar orbits \n(e.g., Yoshii \\& Saio 1979). \nAs a first approximation we therefore neglect the secular dynamical evolution\nof stars and assume that their local density is given by\n\\begin{equation}\n\\rho_*(z) = \\int_{0}^{\\infty}\\frac{\\rho(z,t)}{t_*(z,t)} dt \\ \\ .\n\\end{equation}\n\nA commonly perceived theoretical idea on the formation of galaxies\n(Fall \\& Efstathiou 1980; Blumenthal et al. 1986; Barnes \\& Efstathiou 1987)\nis that their initial state is more or less the virialized isothermal sphere\nwhere baryonic matter relaxes with dark matter. A more recent idea from\ncosmological simulations (Katz 1992) is that spiral galaxies formed through \nthe hierarchical merging of smaller subunits that were disrupted in the \nmerging process. Whereas their dissipationless stellar and dark matter \ncomponents now constitute the visible and dark halos of disk galaxies, the \ngas could dissipate its kinetic energy, spin up and settle into the equatorial\nplane where it formed an initially extended, gaseous protodisk. As we will \ndemonstrate below, the observed exponential vertical disk profiles indicate \nthat this protodisk was able to achieve a local isothermal equilibrium \nstate prior to the onset of star formation. The initial disk temperature \n$T(r)$ then is only a function of the disk radius $r$ but independent of $z$, \nand the initial surface density $\\Sigma (r)$ determines the initial central \ndensity $\\rho (z=0)$ at radius $r$ and its local scale height $h_z (r)$:\n\n\\begin{equation}\n\\rho(z)=\\frac{\\Sigma}{4h_z}{\\rm sech}^2\\left(\\frac{z}{2h_z}\\right) \\;\\;,\n\\;\\;\\;\\;\\;\nh_z=\\frac{k_B T}{2\\pi G\\Sigma\\mu m_H} \\;\\; ,\n\\end{equation}\nwhere $\\mu$ is the mean molecular weight. We use this as a reasonable initial\nsetup in our calculations.\n\n\nThe hydrodynamical equations are solved on an Eulerian staggered grid,\nwith 100 logarithmically spaced grid points in $z$ and an outer edge placed\nat a sufficiently large $z$-distance. The set of differential equations\nis integrated numerically by means of an explicit, finite difference scheme\nwith operator splitting and monotonic transport. This scheme provides\nsecond-order accuracy in space and time (Burkert \\& Hensler 1987).\nThe simplicity of the above equations allows us to make a comprehensive survey \nin the parameter space.\n\n\\section{Discussion of the numerical results}\n\nA number of test calculations have shown that the final stellar $z$-profile\ndepends strongly on the chosen initial distribution of the protodisk gas if \none starts from a non-equilibrium state. On the other hand, an exponential \nstellar disk forms in all cases where we assume the gas to settle into \nisothermal equilibrium, prior to star formation and gas cooling, as long as \nthe star formation time $t_*$ is comparable to the cooling time $t_c$, that \nis, $k\\sim 1$.\n\nIn the limit of rapid star formation ($k\\gg 1$) the initial isothermal\ndistribution of the gas is frozen in the stellar disk which has more or \nless a flat-top $z$-profile near the equatorial plane. In the limit of \nslow star formation ($k\\ll 1$) the gas condenses into the equatorial \nplane towards the energy minimum state, giving rise to a power-law \n$z$-profile in the stellar disk. Exponential profiles are in between \nthese two extremes, and our calculations show that when $k$ lies inside \na preferable range $0.3\\lower.5ex\\hbox{\\ltsima} k\\lower.5ex\\hbox{\\ltsima} 3$, the final stellar $z$-profile \nbecomes exponential, independent of the free parameters $k$, $\\Lambda_0$, \n$a$ and $b$, and also independent of $T$ and $\\Sigma$ of the isothermal \nprotodisk gas. This remarkable result still holds when the isothermal \nequilibrium state is modified to some extent.\n\nIn Figure 1 we show as an example a sequence of models with changing $b$\n(upper panel) or changing $a$ (lower panel) while the other parameters are\nfixed as shown in the panels: $k=1$, $T=10^5$K, $\\Sigma=50M_\\odot$pc$^{-2}$,\nand $\\Lambda_0$ set to give $\\langle t_c\\rangle =10^9$yrs at the half-mass\n$z$-distance of the protodisk. The stellar $z$-profile $\\rho_*(z)$ is almost \nperfectly exponential all the way from $z\\sim0$ to 10$h_z$ spanning about \nfive orders in density, and the scale length becomes systematically smaller \nfor larger $b$ or larger $a$. It is evident that the final scale height \ndepends critically on the adopted values of the parameters and in general \ndiffers from the initial scale height of the gaseous protodisk.\n\nIn Figure 2 we show one of the other sequences of models with decreasing\n$\\Sigma$ from the left to the right, using $a=2$, $b=0$ and the other \nparameters fixed as in Figure 1. $\\Sigma$ is chosen to mimic the situation \nin our own Galaxy with a surface density distribution of \n$\\Sigma=50M_{\\odot}$pc$^{-2}\\times \\exp[-(R-8$kpc)\/4kpc]\nevaluated at galactocentric radial intervals $R=2$kpc$\\times i$ with \n$i=1$ to 5. Note that the profiles have been shifted along the horizontal \naxis. The stellar $z$-profile $\\rho_*(z)$ is almost perfectly exponential\nfor different choices of $\\Sigma$, and the stellar disk flares with\ndecreasing surface density, that is, its thickness increases as \n$h_z\\propto\\Sigma^{-1}$. Such a $\\Sigma$-dependence of $h_z$ would be \nexpected if the stellar system remembers the exponential tail of the \nself-gravitating isothermal disk model which was adopted as an initial \ncondition (see Eq.4). Note however that the final exponential stellar\ndensity distribution extends much further inwards, till $z=0$.\nIncreasing $b$ while keeping $\\Sigma $ and the other parameters constant\ndecreases the scale height according to $h_z(b=1) = 0.6 \\times h_z(b=0)$\nand $h_z(b=2) = 0.4 \\times h_z(b=0)$.\n\nThe disk flaring, when seen edge-on, could largely be masked by projection\neffects. In Figure 3 we show the edge-on projected $z$-profiles $\\Sigma_*(z)$ \nat the same radii as in Figure 2 along the disk plane from the galaxy center. \nThe projection makes the profiles less dependent on $R$. In such a case \nthe inversion for recovering the original density profiles $\\rho_*(z)$ is \ngenerally very unstable. A great accuracy is needed in measuring \n$\\Sigma_*(z)$ over a wide range of $R$ and $z$, in order to detect the disk \nflaring from the observations of edge-on spiral galaxies.\n\nWhile the resulting stellar $z$-profiles are always exponential, the scale\nheight $h_z$ itself depends on the cooling rate $\\Lambda (\\rho ,e)$ through \n$a$ and $b$ and also on the initial state of the protodisk gas through $T$ \nand $\\Sigma$. In practice, however, the deterministic quantities are only \n$T$ and $\\Sigma$ because if these are given the cooling source is virtually \nspecified. Then, $a$ and $b$ are are no longer free parameters. In other \nwords, given that $h_z$ is measured at sufficient accuracy along the disk \nplane, we could constrain $T$ and $\\Sigma$ as a function of radial distance\nfrom the galaxy center. This result can be used in order to explore the\nglobal initial structure of the protodisk from the vertical disk structure \nof an edge-on galaxy observed today.\n\n\\section{Conclusions}\n \nThe presented idea that the star formation timescale $t_*$ is comparable\nto the cooling timescale $t_c$ for the origin of exponential stellar \n$z$-profiles works when the protodisk has nearly reached an equilibrium \nprior to star formation and gas cooling. \n\nUnder a realistic situation the dynamical timescale $t_{dyn}$ of the \nprotodisk may initially not be in balance with the cooling timescale $t_c$. \nIf $t_{dyn}t_c$ otherwise, \nthe gas cools rapidly and condenses into cool and dense clouds. In this \nstage, cloud collisions dissipate kinetic energy and enhance the\nformation of stars. As demonstrated by Burkert et al. (1992), the increase in\nenergy input through supernova explosions will\ncompensate energy dissipation and cooling, \neventually halting the collapse and achieving a quasi-equilibrium state.\nIn either cases, after reaching its \nequilibrium, the protodisk evolves due to the feedback \neffects of star formation, independent of its initial formation history\nand of variations in global physical conditions (e.g., Lin \\& Murray 1992).\n\nGas cooling or energy dissipation induces a slow gravitational settling \nof the protodisk while, at the same time, leading to the formation of stars in \nthe disk. This coupling yields a continuous change of various stellar \ncharacteristics (age, metallicity, color, velocity dispersion, etc) as a \nfunction of the $z$-distance from the disk plane. Although our model is \nnot sufficiently detailed to allow an extensive test against such data, \nthe power of producing an exponential stellar $z$-profile from a vast range \nof physical conditions is very appealing and should be considered as a strong \ncandidate for understanding the universality of exponential $z$-profiles.\n\nBurkert et al. (1992) have performed more detailed simulations of the\ndynamical settling of hot protogalactic gas into the equatorial plane, taking \ninto account star formation and heating and cooling processes in a multiphase \ninterstellar medium. They achieved a good agreement with the observed \nvertical density structure of the Galactic disk in the solar neighborhood. \nThe complexity of their model leads however to a very large parameter space \nwhich cannot be explored in detail due to computational limitations. \nAmong many different processes involved in their physical model, the process \nof crucial importance is that the rate of star formation is \nadjusted sooner or later to balance with the local cooling rate by means of\nthe self-regulated star formation mechanism (Cox 1983; Franco \\& Cox 1983). \nIt is not clear from their models, which process is most\nimportant in leading to exponential $z$-profiles.\nOur calculations demonstrate that an ideal condition for such a profile\nis $t_*\\sim t_c$, independent of the details of heating and cooling.\n\n\\subsection*{Acknowledgments}\n\nWe thank K. Freeman and H. Saio for invaluable comments on the subject\ndiscussed in this paper. A.B. acknowledges the financial support from \nthe Program of Human Capital Mobility, Grant Number ERBCHGECT920009 and \nof CESCA Consortium and the hospitality of the Centre d'Estudis Avancats \nde Blanes. Y.Y. acknowledges the financial support from the Yamada Science\nFoundation, the German Academic Exchange Service (DAAD), and also Theoretical\nAstrophysics Center, Denmark, under which this work was performed. \nThis work has been supported in part by the Grant-in-Aid for COE Research \n(07CE2002). All calculations have been performed on the Cray YMP 4\/64 of \nthe Rechenzentrum Garching.\n\n\\newpage\n\n\\begin{center}\n{\\small REFERENCES}\n\\end{center}\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ Aoki, T. E., Hiromoto, N., Takami, H., \\& Okamura, S. 1991, PASJ,\n43, 755\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ Bahcall, J. N., \\& Soneira, R. 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C., \\& Hyland, A.R. 1989, ApJ, 337, 163\n\n\\clearpage\n\n\\begin{center}\n{\\small FIGURE CAPTIONS}\n\\end{center}\n\n\\vspace{1.0cm}\n\n{\\normalsize F{\\footnotesize IG}.}\n1.---The final stellar $z$-profile $\\rho_*(z)$ for different values of the\npower index $b$ (upper panel) or $a$ (lower panel) for the cooling rate defined\nas $\\Lambda = \\Lambda_0 \\rho^a e^b$, with other parameters fixed in the model,\nas shown in the figure (for details see text).\nThe $z$-profiles (thick lines) originate from the same initial\ndistribution\nof the protodisk gas (dotted line).\n\n\\vspace{1.0cm}\n\n{\\normalsize F{\\footnotesize IG}.}\n2.---The final stellar $z$-profile $\\rho_*(z)$ for different values\nof the initial surface mass density of the protodisk $\\Sigma$ (face-on\nprojected), with other parameters fixed in the model (for details see text).\nThe five $z$-profiles from the left to the right show a sequence of\ndecreasing $\\Sigma$, according to the formula appropriate to our own Galaxy:\n$\\Sigma=50M_{\\odot}$pc$^{-2}\\times \\exp[-(R-8$kpc)\/4kpc] evaluated at\ngalactrocentric radial intervals $R=2$kpc$\\times i$ with $i=1$ to 5.\nNote that the profiles have been shifted along the horizontal axis.\n\n\n\\vspace{1.0cm}\n\n{\\normalsize F{\\footnotesize IG}.}\n3.---The edge-on projected $z$-profiles $\\Sigma_*(z)$ at the same radii as\nin Figure 2 along the disk plane from the galaxy center.\nNote that the profiles have been shifted along the horizontal axis.\n\n\\end{document}\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\subsection{Astronomical motivation}\\label{sec:intro} \nMillimeter and submillimeter wavelength continuum emission \nis a powerful probe of the warm and cool dust in the Universe.\nFor temperatures below $\\sim$40\\,K, the peak of the thermal continuum emission is at wavelengths longer than 100\\,\\micron \n(or at frequencies lower than 3\\,THz), i.e.~in the far-infrared and (sub)millimeter\\footnote{Hereafter we will use the term {\\it (sub)millimeter} when referring, in general, to the millimeter and submillimeter wavelengths range;\nwe will use {\\it submillimeter} when strictly referring to wavelengths from one millimeter down to the far-infrared.} range.\nA number of {\\it atmospheric windows} between 200\\,GHz and 1\\,THz make ground-based observations possible over a large part of\nthis range from high-altitude, dry sites.\n\nSpecifically, thermal dust emission is well described by a gray body spectrum,\nwith the measured flux density $S_\\nu$ at frequency $\\nu$ expressed as:\n\\[ \nS_\\nu = \\Omega_{S'} ~ (1-e^{-\\tau}) ~ \\frac{2h}{c^2}\n\\frac{ \\nu^3 } {e^{h\\nu \/ kT} - 1}\n\\]\nwhere $h$ and $k$ are Planck's and Boltzmann's constants respectively,\n$c$ is the speed of light, $\\Omega_{S'}$ is the apparent source solid angle\n(the size of the physical source convolved with the telescope beam)\nand $\\tau$ is the optical depth, which varies with frequency.\nIn the (sub)millimeter range, the emission is almost always optically thin with $\\tau \\propto N \\nu^\\beta$,\nwhere $\\beta$ is in the range 1--2 typically\n(see, e.g.,~the Appendix of \\citealt{Mezger1990} or \\citealt{Beuther2002} for a more thorough discussion).\nHere, $N$ is the number of dust particles (or number of nucleons assuming a given dust-to-gas\nrelation) in the telescope beam. Accordingly, the (sub)millimeter flux\ncan be converted into dust\/gas masses, when the temperature $T$ is\nassumed or constrained via additional far-infrared measurements.\n\nSuch mass determination is one of the core issues of (sub)millimeter\nphotometry. We would like to illustrate its paramount importance with\na few examples: (sub)millimeter wavelengths mapping of low-mass\nstar-forming regions in molecular clouds have determined the dense\ncore mass spectrum, (in nearby regions) down to sub-stellar masses,\nand investigated its relationship to the Initial Mass Function\n\\citep{Motte1998}. These studies can be extended to high-mass\nstar-forming regions \\citep[e.g.~][]{Motte2007,\nJohnstone2006}. Because of the larger distances to rarer, high-mass\nembedded objects, even relatively shallow surveys are capable of\ndetecting pre-stellar cluster clumps with a few hundreds of solar\nmasses of material at distances as far as the Galactic center. Masses\nand observed sizes yield radial density distribution profiles for\nprotostellar cores that can be compared with theoretical models\n\\citep{Beuther2002}.\n\n(Sub)millimeter continuum emission is also a remarkable tool for the\nstudy of the distant Universe. The thermal dust emission in active \ngalaxies is typically fueled by short-lived, high-mass stars, therefore\nthe far-infrared luminosity provides a snapshot of the current level of star-formation activity.\nDeep (sub)millimeter observations in\nthe Hubble Deep Field, with the Submillimeter Common User Bolometer\nArray \\citep[SCUBA,][]{1999MNRAS.303..659H} on the James Clerk Maxwell\nTelescope, attracted considerable attention with the detection of a\nfew sources without optical or near-infrared counterpart. Sensitive\nmeasurements found dust in high redshift sources and even in some of\nthe farthest known objects in the Universe \\citep[see, e.g.,][]{Carilli2001, Bertoldi2003, Wang2008},\nrevealing star-formation rates (SFRs) that are hundreds of times higher than in\nthe Milky Way today. These detections are possible because of the so-called\n{\\it negative K-correction} first discussed by \\citet{Blain1993}: the\nwarm dust in galaxies is typically characterized by temperatures\naround 30--60\\,K. Its thermal emission dominates the spectral energy\ndistribution (SED) of luminous and ultra-luminous infrared galaxies\n(LIRGs and ULIRGs), which have maxima between 3 and 6\\,THz as a\nresult. Because of the expansion of the Universe, the peak of the\nemission shifts toward the lower frequencies with increasing\ncosmological distance, thus counteracting the dimming and benefitting \ndetection at (sub)millimeter wavelength. Consequently,\nflux-limited surveys near millimeter wavelengths yield flat or nearly\nflat luminosity selection over much of the volume of the Universe\n\\citep{Blain02}. As such, (sub)millimeter wavelengths allow unbiased\nstudies of the luminosity evolution and, therefore, of the star-formation\nhistory of galaxies over cosmological time-scales.\n\\subsection{Bolometer arrays for (sub)millimeter astronomy}\n\\begin{figure}[t]\n\\centering \n\\includegraphics[width=.96\\linewidth, bb=20 20 1364 1316, clip]{1454fig01light.eps}\n\\caption\n{\nWiring side of a naked LABOCA array. Each light-green square is a bolometer.\n}\n\\label{fig:nakedarray}\n\\end{figure}\nThe (sub)millimeter dust emission is unfortunately intrinsically weak and\nthe wish to measure it pushed the development of detectors with the best\npossible sensitivity, namely bolometers \\citep{Low1961, Mather1984}.\nMoreover, the desirability of mapping large areas of the sky,\nmotivated the development of detector {\\it arrays}.\nConsequently, the last 10 years have seen an\nincreasing effort in the development of bolometer arrays. In such\ninstruments, a number of composite bolometers work side by side in the\nfocal plane, offering simultaneous multi-beam coverage. Since the\narrival of the first arrays, developed in the early '90s and\nconsisting of just 7 elements, we are witnessing a rapid maturing of\ntechnology, reaching hundreds to a few thousand elements today, and\nthe prospect of even larger bolometer arrays in the future.\nThe Large APEX Bolometer Camera, LABOCA, described in this article,\nis a new bolometric receiver array for the Atacama Pathfinder Experiment 12\\,m telescope,\nAPEX\\footnote{APEX is a collaborative effort between the\nMax-Planck-Institut f\\\"ur Radioastronomie of Bonn (MPIfR,\\,50\\%), the\nEuropean Southern Observatory (ESO,\\,27\\%) and the Onsala Space\nObservatory (OSO,\\,23\\%)}. It is the most ambitious camera in a long\nline of developments of the MPIfR bolometer group, which has delivered\ninstruments of increasing complexity to the IRAM 30 m telescope\n\\citep{Kreysa1990}: single beam receivers were supplanted by a\n7-element system \\citep{Kreysa1999}, which eventually gave way to the\nMAx-Planck Millimeter BOlometer (MAMBO) array, whose initial 37 beams\nhave grown to 117 in the latest incarnation \\citep{2002AIPC..616..262K}.\nThe group also built the 37-element 1.2\\,mm SEST Imaging Bolometer Array (SIMBA)\nfor the Swedish\/ESO Submillimeter telescope \\citep{Nyman2001}\nand the 19-element 870\\,\\micron for the 10\\,m Heinrich-Hertz Telescope\n\\citep[a.k.a. SMTO,][]{1990SPIE.1235..503M}.\n\nThe main obstacle to observations at these wavelengths is posed by Earth's atmosphere, which is\nseen as a bright emitting screen by a continuum total power detector, as LABOCA's bolometers are.\nThis is largely due to the emission of the water vapor present in the atmosphere with only small\ncontributions from other components, like ozone. Besides, the atmosphere is a turbulent thermodynamic\nsystem and the amount of water vapor along the line of sight can change quickly, giving rise to instabilities\nof emission and transmission, called {\\it sky noise}. Observations from ground based telescopes have to go\nthrough that screen, therefore requiring techniques to minimize those effects.\n\nThe technique most widely used is to operate a switching device,\nusually a chopping secondary mirror (hereafter called {\\it wobbler})\nto observe alternatively the source and a blank sky area close to it,\nat a frequency higher than the variability scale of the sky noise.\nThis method, originally introduced for observations with single pixel detectors, is today also used with arrays of bolometers.\nAlthough it can be very efficient to reduce the atmospheric disturbances during observations,\nit presents some disadvantages: among others, the wobbler is usually slow (1 or 2 Hz) posing a\nstrong limitation to the possible scanning speed.\n\nLABOCA has been specifically designed to work without a wobbler and using a different technique,\nwhich works particularly well when using an array of detectors, to remove the atmospheric contribution.\nThis technique, called {\\it fast scanning} \\citep{2001A&A...379..735R}, is based on the idea that, when observing\nwith an array, the bolometers composing the array look simultaneously at different points in the sky,\ntherefore chopping is no more needed.\nA modulation of the signal, still required to identify the astronomical source through the atmospheric\nemission, is produced by scanning with the telescope across the source.\nThe atmospheric contribution (as well as part of the instrumental noise) will be strongly correlated in\nall bolometers and a post-detection analysis of the correlation across the array will \nmake it possible to extract the signals of astronomical interest from the atmospheric foregrounds.\nMoreover, the post-detection bandwidth depends on the scanning speed, therefore relatively high\nscanning speeds are ideal \\citep[see also][]{Kovacs2008}.\n\nThe APEX telescope \\citep{2006A&A...454L..13G}, as the name implies,\nserves as a pathfinder for the future large-scale (sub)millimeter\nwavelength and (far)infrared missions, namely the Atacama Large\nMillimeter Array (ALMA), the Herschel Space Observatory and the\nStratospheric Observatory for Infrared Astronomy (SOFIA). Its\npathfinder character is on the one hand defined by exploring\nwavelength windows that have been poorly studied before, with\nacceptable atmospheric transmission at the 5100\\,m altitude site. On\nthe other hand, and more importantly, it can perform large area\nmapping to identify interesting sources for ALMA follow-up studies at\nhigher angular resolution. Moreover, APEX produces images of both\ncontinuum (with LABOCA) and line emission with angular resolutions\nthat neither Herschel's nor SOFIA's smaller telescopes (with diameters\nof 3.5 and 2.5\\,m respectively) can match. This provides a critical\nadvantage to APEX for imaging dust and line emission at high frequencies. \n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=.90\\linewidth,bb=22 22 699 535, clip]{1454fig02light.eps}\n\\caption\n{\nScheme of the infrastructure of LABOCA. The instrument is located in the Cassegrain cabin (dashed red line)\nof APEX, remote operation is a requirement and all the communication goes through\nthe local area network (magenta arrows) and a direct Ethernet link between {\\it backend} and {\\it bridge} computers.\nThe black lines show the flow of the bolometer signals, the orange ones show the configuration and monitoring communication.\n}\n\\label{fig:infrastructure}\n\\end{figure*}\n\\subsection{Instrument overview}\\label{sec:overview}\nLABOCA is an array of bolometers, operated in total-power mode, specifically designed \nfor fast mapping of large areas of sky at moderate resolution and with high \nsensitivity and was commissioned in May 2007 as facility instrument on APEX.\nIt is a very complex system, composed of parts that originate in a variety of\nfields of technology, in particular optics, high vacuum,\nlow temperature cryogenics, digital electronics, computer hardware and\nsoftware, and others. A general view of the infrastructure is shown\nin the block diagram of Fig.~\\ref{fig:infrastructure}.\nThe heart of LABOCA is its detector array made of 295 semiconducting composite bolometers\n(see Fig.~\\ref{fig:nakedarray}, Fig.~\\ref{fig:arraypics}).\nA description of the detector array design and manufacture is provided in \\S\\ref{sec:detector}.\n\nThe bolometer array is mounted in a cryostat, which uses liquid\nnitrogen and liquid helium on a closed cycle double-stage sorption\ncooler to reach an operation temperature of $\\sim$285\\,mK. The\ncryogenic system is discussed in \\S\\ref{sec:cryo}.\n\nA set of cold filters, mounted on the liquid nitrogen and liquid\nhelium shields, define the spectral passband, centered at a wavelength\nof 870\\,\\micron and about 150\\,\\micron wide (see\nFig.~\\ref{fig:passband}). A monolithic array of conical horn\nantennas, placed in front of the bolometer wafer, concentrates the\nradiation onto the individual bolometers. The filters and the horn\narray are presented in \\S\\ref{sec:coptics}.\n\nThe cryostat is located in the Cassegrain cabin of the APEX telescope\n(see Fig.~\\ref{fig:receiver}) and the optical coupling to the\ntelescope is provided by an optical system made of a series of metal\nmirrors and a lens which forms the cryostat window. The complex\noptics layout, manufacture and installation at the telescope in\ndescribed in \\S\\ref{sec:optics}.\n\nThe bolometer signals are routed through low noise, unity gain\nJunction Field Effect Transistor (JFET) amplifiers and to the outside\nof the cryostat along flexible flat cables. Upon exiting the\ncryostat, the signals pass to room temperature low noise amplifiers\nand electronics also providing the AC current for biasing the\nbolometers and performing real time demodulation of the signals. The\nsignals are then digitized over 16 bits by 4 data acquisition boards\nproviding 80 analog inputs each, mounted in the {\\it backend\ncomputer}. The backend software provides an interface to the\ntelescope's control software, used to set up the hardware, and a data\nserver for the data output. The acquired data are then digitally\nfiltered and downsampled to a lower rate in real time by the {\\it\nbridge computer} and finally stored in MB-FITS format\n\\citep{2006A&A...454L..25M} by the FITS writer embedded in the\ntelescope's control software. Another computer, the {\\it frontend\ncomputer}, is devoted to monitor and control most of the\nelectronics embedded into the receiver (e.g.~monitoring of all the\ntemperature stages, controlling of the sorption cooler, calibration\nunit) and provides an interface to the APEX control software, allowing\nremote operation of the system. Discussions of the cold and warm\nreadout electronics, plus the signal processing, are found in\n\\S\\ref{sec:jfet} and \\S\\ref{sec:electronics}, respectively.\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=.92\\linewidth, bb= 20 20 1147 846,clip]{1454fig03light.eps}\n\\caption\n{\nOverview of the optics. On the left, the APEX telescope.\nOn the right, a zoom on the tertiary optics installed in the Cassegrain cabin.\nSee also Fig.~\\ref{fig:receiver}.\n}\n\\label{fig:opticsall}\n\\end{figure*}\n\nIn \\S\\ref{sec:obsmodes} we describe the\nsophisticated observing techniques used with LABOCA,\nsome of them newly developed. \nThe instrument performance on the sky is described in \\S\\ref{sec:performance},\nalong with information on sensitivity, beam shape and noise behavior.\n\nThe reduction of the data is performed using a new data reduction\nsoftware included in the delivery of LABOCA as facility instrument,\nthe BoA (Bolometer array data Analysis) data reduction software package.\nAn account of on-line and off-line data reduction is given in \\S\\ref{sec:redu}.\n\nSome of the exciting science results already obtained with LABOCA or\nexpected in the near future are outlined in \\S\\ref{sec:science}. Our\nplans for LABOCA's future are briefly presented in \\S\\ref{sec:future}.\n\\section{Tertiary optics}\\label{sec:optics}\n\\subsection{Design and optimization} \nThe very restricted space in the Cassegrain cabin of\nAPEX and a common first mirror (M3) with the APEX SZ Camera\n\\citep[ASZCa,][]{2003NewAR..47..933S} introduced many boundary\nconditions into the optical design. Eventually, with the help of the\nZEMAX\\footnote{\\tiny{\\url{http:\/\/www.zemax.com\/}}} optical design program, a\nsatisfactory solution was found, featuring three aspherical off-axis\nmirrors (M3, M5, M7), two plane mirrors (M4, M6) and an aspherical\nlens acting as the entrance window of the cryostat (see\nFig.~\\ref{fig:opticsall}). Meeting the spatial constraints, without\nsacrificing optical quality, is facilitated considerably by the\naddition of plane mirrors.\nThe design of the optics was made at the MPIfR in coordination with N.\\ Halverson\\footnote{Formerly\nat University of California at Berkeley, now at University of\nColorado, Boulder, USA} with respect to sharing the large M3 mirror\nwith the ASZCa experiment.\n\nThe maximum possible field diameter of APEX, as limited by the\ndiameter of the Cassegrain hole of the telescope's primary, is about\n0.5 degrees. LABOCA, with its 295 close-packed fully efficient horns,\ncovers an almost circular field of view (hereafter FoV) of about 0.2\ndegrees in diameter (or about 100 square arcminutes). The task of the\ntertiary optics is to transform the f-ratio from f\/8 at the Cassegrain\nfocus to f\/1.5 at the horn array, while correcting the aberrations\nover the whole FoV of LABOCA under the constraint of parallel output\nbeams. The final design is diffraction limited even for 350\\,\\micron \nwavelength, the Strehl ratio is better than 0.994 and the maximum\ndistortion at the focal plane is less than 10\\% over the entire FoV\n(see also Fig.~\\ref{fig:beams}).\n\\begin{figure}[t] \n\\centering \n\\includegraphics[width=.95\\linewidth,bb=20 20 575 436, clip]{1454fig04toplight.eps}\n\\includegraphics[width=.95\\linewidth, bb=20 20 560 740,clip]{1454fig04bottomlight.eps} \n\\caption\n{\n{\\it Top:}~The mirrors affixed to the floor of the Cassegrain cabin of APEX. Left to right: M7, M5, M3 and\none of the mirrors of the ASZCa experiment. In this picture, mirror M3 is positioned for ASZCa.\n{\\it Bottom:}~The receiver, M3 in the position for LABOCA, M5, M6 and M7.\nSee also Fig.~\\ref{fig:opticsall}.\n}\n\\label{fig:receiver}\n\\end{figure}\n\\subsection{Manufacture} \nMirror M3, an off-axis paraboloid, has been\nmanufactured by a machine shop at the Lawrence Berkeley National\nLaboratory (LBNL, Berkeley, CA, USA) and is common to both LABOCA and the ASZCa\nexperiment. M3 has a diameter of 1.6\\,m and a surface accuracy of\n18\\,\\micron rms but LABOCA uses only the inner 80\\,cm disk. It is\nattached to a bearing on the floor of the Cassegrain cabin, aligned to\nthe optical axis of APEX. Operators can manually rotate the mirror in\norder to direct the telescope beam to LABOCA or to ASZCa alternatively\n(see Fig.~\\ref{fig:receiver}).\n\nMirrors M4 and M6 are flat, have a diameter of 42\\,cm and 26\\,cm,\nrespectively (manufactured by Kugler\\footnote{\\tiny{\\url{http:\/\/www.kugler-precision.com\/}}}, Salem, Germany).\nThey are of optical quality and are both affixed to the ceiling of the cabin.\nIn a near future, mirror M6 will be replaced by the reflection-type half-wave\nplate of the PolKa polarimeter \\citep{2004A&A...422..751S}.\n\nMirrors M5 and M7 are off-axis aspherics, both 50\\,cm in diameter, and\nare affixed to the floor of the cabin. They have been designed and\nmanufactured at the MPIfR and have a surface accuracy of 7 and\n5\\,\\micron rms respectively.\n\\subsection{Installation and alignment} \nAll the mirrors of LABOCA\n(with exception of M3) and the receiver itself are mounted on hexapod positioners (see Fig.~\\ref{fig:receiver})\nprovided by VERTEX Antennentechnik\\footnote{\\tiny{\\url{http:\/\/www.vertexant.de\/}}} (Duisburg, Germany).\nEach hexapod is made of an octahedral assembly of struts and has six degrees of freedom (x, y, z, pitch,\nroll and yaw). The lengths of the six independent legs can be changed\nto position and orient the platform on which the mirror is mounted.\nVERTEX provided a software for calculating the required leg extensions\nfor a given position and orientation of the platforms.\nA first geometrical alignment was performed during the first week of September\n2006, using a double-beam laser on the optical axis of the telescope\nand plane replacement mirrors in place of the two active mirrors M5\nand M7. The alignment has been checked using the bolometers and hot\ntargets (made of absorbing\\footnote{ECCOSORB AN, Emerson \\& Cuming, Rundolph, MA, USA, \\tiny{\\url{http:\/\/www.eccosorb.com}}} material)\nat different places along the beam, starting at the focal plane and\nfollowing the path through all the reflections up to the receiver's\nwindow. The alignment has been furthermore verified and improved in\nFebruary 2008.\n\\section{Cryogenics}\\label{sec:cryo}\n\\subsection{Cryostat} \nThe bolometer array of LABOCA is designed to be operated at a temperature\nlower than 300\\,mK. This temperature is provided by a cryogenic\nsystem made of a wet cryostat, using liquid nitrogen and liquid\nhelium, in combination with a two-stage sorption cooler. A commercial\n8-inch cryostat, built by Infrared Labs\\footnote{\\tiny{\\url{http:\/\/www.irlabs.com\/}}} (Tucson, AZ, USA), has been\ncustomized at the MPIfR to accommodate the double-stage sorption\ncooler, the bolometer array, cold optics and cold electronics. A high\nvacuum in the cryostat is provided by an integrated turbomolecular\npump backed by a diaphragm pump. Operational vacuum is reached in one\nsingle day of pumping.\n\nThe cryostat incorporates a 3-liter reservoir of liquid nitrogen and a\n5-liter reservoir of liquid helium. After producing high-vacuum\n($\\sim$10$^{-6}$\\,mbar), the cryostat is filled with the liquid\ncryogens. The liquid nitrogen is used to provide thermal shielding at\n77\\,K in our labs in Bonn (standard air pressure, 1013\\,mbar) and at\n73.5\\,K at APEX (5107\\,m above the sea level) where the air pressure\nis almost one half of the standard one (about 540\\,mbar).\n\nThe liquid helium provides a thermal shielding at 4.2\\,K at standard\npressure and 3.7\\,K at the APEX site. To keep it operational, the\nsystem must be refilled once per day. The refilling operation\nrequires about 20~minutes.\n\\subsection{Sorption cooler}\nThe cryostat incorporates a commercial two-stage closed-cycle sorption\ncooler, model SoCool \\citep{2002AIPC..613.1233D} manufactured by\nAir-Liquide\\footnote{\\tiny{\\url{http:\/\/www.airliquide.com\/}}} (Sassenage,\nFrance). In this device, a \\element[][4]{He} sorption cooler is used to\nliquefy \\element[][3]{He} gas in the adjacent, thermally coupled, \\element[][3]{He}\ncooler. The condensed liquid \\element[][3]{He} is then sorption pumped to\nreach temperatures as low as 250\\,mK, in the absence of a thermal\nload. Therefore, the double stage design makes it possible to cool the\nbolometer array down to a temperature lower than 300\\,mK starting from the\ntemperature of the liquid helium bath at atmospheric pressure. This\nmakes the maintenance of the system much simpler than that of other\nsystems, where pumping on the liquid helium bath is required. The two\nsorption coolers are closed systems, which means they do not require\nany refilling of gas and can be operated from the outside of the\ncryostat, simply by applying electrical power.\n\nTo keep the bolometers at operation temperature, the sorption cooler\nneeds to be recycled. The recycling is done by application of a\nsequence of voltages to the electric lines connected to thermal\nswitches and heaters integrated in the sorption cooler. A typical\nrecycling procedure requires about two hours and can be done manually\nor in a fully automatic way controlled by the frontend computer (see\nSect.~\\ref{sec:fepc}). At the end of the recycling process, both\ngases, \\element[][4]{He} and \\element[][3]{He}, have been liquefied and the controlled\nevaporation of the two liquids provides a stable temperature for many\nhours. After the recycling of the sorption cooler, the bolometer\narray reaches 285\\,mK. The hold time of the cooler, usually between\n10 and 12 hours, strongly depends on the parameters used during the\nrecycling procedure.\nThe end temperature is a function of elevation and can be affected by telescope movements,\nleading to temperature fluctuations $\\la500\\,$\\microK within one scan,\nduring regular observations, and $\\la3\\,$mK for wide elevation turns\n(e.g. during skydips, see Sect.~\\ref{sec:skydips}).\n\\subsection{Temperature monitor}\\label{sec:thermonitor}\nThe cryostat of LABOCA incorporates 8\nthermometers to measure the temperature at the different stages:\nliquid nitrogen, liquid helium, the two sorption pumps, the two\nthermal switches, evaporator of the \\element[][4]{He} and evaporator of the\n\\element[][3]{He}. Two LS218\\footnote{\\tiny{\\url{http:\/\/www.lakeshore.com\/temp\/mn\/218po.html}}}\ndevices (Lake Shore Inc., Westerville, OH, USA) are used to monitor the thermometers\nand to apply the individual temperature calibrations in real-time. \nThe temperature of the \\element[][3]{He} stage is measured with higher accuracy with the use of a\nresistance bridge AVS-47\\footnote{\\tiny{\\url{http:\/\/www.picowatt.fi\/avs47\/avs47.html}}}\n(Picowatt, Vantaa, Finland), with an error of $\\pm$5\\,\\microK.\nControl and monitor of the cryogenic\nequipment can be done remotely via the frontend computer (see\nSect.~\\ref{sec:fepc}).\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.98\\linewidth, bb=124 93 764 523, clip]{1454fig05light.eps}\n\\caption\n{\nSpectral response of LABOCA, relative to the maximum.\nThe central frequency is 345\\,GHz, the portion with 50\\% or more\ntransmission is between 313 and 372\\,GHz.\n}\n\\label{fig:passband}\n\\end{figure}\n\\section{Cold optics}\\label{sec:coptics}\n\\subsection{Passband definition}\\label{sec:passband}\nInside the cryostat, a set of cold filters, mounted on\nthe liquid nitrogen and liquid helium shields define the spectral\npassband, centered at a wavelength of 870\\,\\micron (345\\,GHz) and about\n150\\,\\micron (60\\,GHz) wide (see Fig.~\\ref{fig:passband}). The filters\nhave been designed and manufactured at MPIfR in collaboration with the\ngroup of V.~Hansen (Theoretische Elektrotechnik\\footnote{\\tiny{\\url{http:\/\/www.tet.uni-wuppertal.de\/}}}, Bergische\nUniversit\\\"at Wuppertal, Germany) who provided theoretical support and\nelectromagnetic simulations. The passband is formed by an\ninterference filter made of inductive and capacitive meshes embedded\nin polypropylene. The low frequency edge of the band is defined by\nthe cut-off of the cylindrical waveguide of each horn antenna (see\nalso Sect.~\\ref{sec:horns}). A freestanding inductive mesh behind the\nwindow-lens provides shielding against radio interference.\n\\subsection{Horn array}\\label{sec:horns} \nA monolithic array of conical\nhorn antennas, placed in front of the bolometer wafer, concentrates\nthe radiation onto the bolometers. 295 conical horns have been\nmachined into a single aluminum block by the MPIfR machine shop. In\ncombination with the tertiary optics, the horn antennas are optimized\nfor coupling to the telescope's main beam at a wavelength of\n870\\,\\micron . The grid constant of the hexagonal array is\n4.00\\,mm. Each horn antenna feeds into a circular wave guide with a\ndiameter of 0.54\\,mm, acting as a high-pass filter.\n\\section{Detector}\\label{sec:detector}\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=.32\\linewidth, bb= 0 0 200 200, clip]{1454fig06leftlight.eps}\n\\includegraphics[width=.32\\linewidth, bb= 0 0 200 200, clip]{1454fig06centerlight.eps}\n\\includegraphics[width=.32\\linewidth, bb= 0 0 200 200, clip]{1454fig06rightlight.eps}\n\\caption\n{\nPictures of the bolometer array of LABOCA.\n{\\it Left:}~A detail of the array mounted in its copper ring. Some bonding wires are visible.\n{\\it Center:}~The side of the array where the bolometer cavities are etched in the silicon wafer.\n{\\it Right:}~Wiring side of the array. The thermistors are visible as small cubes on the membranes.\nOne broken membrane is visible on the top left corner. See also Fig.~\\ref{fig:nakedarray}.\n}\n\\label{fig:arraypics}\n\\end{figure*}\n\\subsection{Array design and manufacture} \\label{sec:array}\nThe bolometer array of LABOCA is made of 295 composite bolometers arranged in an hexagonal\nlayout consisting of a center channel and 9 concentric hexagons (see Fig.~\\ref{fig:nakedarray}\nand Fig.~\\ref{fig:arraypics}). The array is manufactured on a 4-inch silicon wafer coated on both sides with\na silicon-nitride film by thermal chemical vapor deposition. On one side of the wafer,\n295 squares are structured into the silicon-nitride film used as a mask for the alkaline KOH\netching of the silicon, producing freestanding, unstructured silicon-nitride membranes, only\n400\\,nm thick (see the center picture in Fig.~\\ref{fig:arraypics}).\nOn the other side, the wiring is created by microlithography of\nniobium and gold thin metal layers. The bolometer array is mounted\ninside a gold-coated copper ring and is supported by about 360 gold\nbonding thin wires (see Fig.~\\ref{fig:arraypics},~left), providing the\nrequired electrical and thermal connection. This copper ring\nalso serves as a mount for the backshort reflector, at $\\lambda$\/4 distance from\nthe array, and 12 printed circuit boards hosting the load resistors\nand the first electronic circuitry (see Sect.~\\ref{sec:jfet}).\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.94\\linewidth, bb=0 0 233 173, clip]{1454fig07light.eps}\n\\caption\n{\nWiring side of one bolometer of LABOCA, seen under a microscope.\nThe membrane is the green colored area. The black box is the NTD thermistor.\nThe wiring is made of gold (yellow) and niobium (gray) thin metal layers.\nSee also Fig. \\ref{fig:arraypics}.\n}\n\\label{fig:bolo}\n\\end{figure}\n\\subsection{Bolometer design and manufacture} \nThe description of a composite bolometer can be simplified as the combination of two elements:\nan extremely sensitive temperature sensor, called thermistor, and a radiation absorber. \nIn the bolometers of LABOCA, the absorbing element is made of a thin film of titanium deposited on\nthe unstructured silicon-nitride membranes. LABOCA uses neutron-transmutation doped germanium semiconducting chips\n\\cite[NTD,][]{1982STIN...8328412H} as thermistors. The NTD thermistors are made from ultra-pure germanium doped by\nneutron-transmutation in a nuclear reactor. The thermistors, needed to measure the temperature of the absorber,\nare soldered on the bolometer membranes to gold pads connected to the outer edge of the silicon wafer through a\npair of patterned niobium wires, superconducting at the operation temperature (see Fig.~\\ref{fig:bolo} and Fig. \\ref{fig:arraypics}, right).\nThe soldering of the thermistors was the only manual step in the manufacture of the bolometers.\nLABOCA has so-called {\\it flatpack} NTD thermistors, which have two ion\nimplanted and metalized contacts on one side of the thermistor block.\nThe chips are optimized to work at a temperature lower than 300\\,mK,\nwhere they show an electric impedance in the range of 1 to 10\\,MOhms.\n\\section{Cold electronics}\\label{sec:jfet} \n\\subsection{Bias resistors}\nGiven the high impedance of the NTD thermistors, the electric scheme\nof the first bolometer circuitry requires very high impedance load\nresistors, which are needed to current bias the bolometers. These\nbias resistors are 312 identical 30\\,MOhm chips, made of a nichrome\nthin film deposited on silicon substrate (model MSHR-4 produced by\nMini-Systems Inc.\\footnote{\\tiny{\\url{http:\/\/www.mini-systemsinc.com\/}}},\nN. Attleboro, MA, USA) mounted on 12 identical printed circuit boards, to\nform 12 groups of 26 resistors. This configuration is reflected in the\nfollowing distribution of the bolometer signals. The circuit boards\nare mounted on the same copper ring which holds the bolometer array\n(see Sect.~\\ref{sec:array})\nand electrically connected to the bolometers through miniature RF\nfilters.\n\n\\subsection{Junction Field Effect Transistors (JFETs) source followers} \nThe high impedance of the bolometers makes the system\nsensitive to microphonic noise pickup, therefore JFETs (Toshiba 2SK369)\nare used as source followers in order to decrease the impedance of the\nelectric lines down to a few kOhms before they reach the high gain\namplification units at room temperature. Following the wiring scheme\nof the bias resistors, the JFETs are also in groups of 26 soldered\nonto 12 printed circuit boards, electrically connected to the\ncorresponding bias resistors by 12 flat cables made of manganin traces embedded in\nKapton\\footnote{DuPont, \\tiny{\\url{http:\/\/www2.dupont.com\/Kapton\/en_US\/}}}\n(manufactured by VAAS Leiterplattentechnologie\\footnote{\\tiny{\\url{http:\/\/www.vaas-lt.de\/}}},\nSchw\\\"abisch-Gm\\\"und, Germany), thermally shunted to the liquid helium tank.\nThe 12 JFET boards are assembled in groups of\nthree into four gold-coated copper boxes, thermally connected to the\nliquid nitrogen tank. Inside each box, during regular operation, the\n78 JFETs are self-heated to a temperature of about 110\\,K, where they\nshow a minimum of their intrinsic noise. Through the connections in\nthe JFET boxes, the wiring scheme of 12 groups of 26 channels is\ntranslated to a new scheme of 4 groups of 80 channels.\n\\section{Warm electronics and signal processing}\\label{sec:electronics}\n\\subsection{Amplifiers}\\label{sec:ampli}\nThe 312 channels exiting the\ncryostat of LABOCA are distributed to 4 identical, custom made,\namplification units, providing 80 channels each. Of these, 295 are\nbolometers, 17 are connected to 1\\,MOhm resistors mounted on the\nbolometer ring (used for technical purposes, like noise monitoring and\ncalibrations) and the remaining 8 are not connected. Each\namplification unit is made of 16 identical printed circuit boards and\neach board provides 5 low noise, high gain amplifiers. Each unit also\nincludes a low noise battery used to generate the bias voltage and the\ncircuitry to produce the AC biasing and perform real time demodulation\nof the 320 signals. The AC bias reference frequency is not internally\ngenerated but is provided from the outside, thus allowing\nsynchronization of the biasing to an external frequency source.\n\nEach amplification unit is equipped with a digital interface controlled by a microprocessor programmed to provide remote control of\nthe amplification gain and of the DC offset removal procedure. This is required because LABOCA is a total power receiver and the\nsignals carry a floating DC offset which could exceed the dynamic range of the data acquisition system.\nTo avoid saturation, therefore, the DC offsets are measured and subtracted from the signals at the beginning of every integration.\nThe values of the 320 offsets are temporarily stored in a local memory and, at the end of the observation, are written into the\ncorresponding data file for use in the data reduction process.\nThe digital lines use the I$^2$C protocol\\footnote{Inter-Integrated Circuit, a serial bus to connect hardware devices.}\nand are accessible remotely via the local network through I$^2$C-to-RS232\\footnote{Recommended Standard 232, a standard for serial\nbinary data communication.} interfaces controlled by the frontend computer (see Sect.~\\ref{sec:fepc}).\nThe amplification gain can be set in the range from 270 to 17280.\n\\subsection{Data acquisition}\\label{sec:daq}\nThe 320 output signals from the 4 amplification units are digitized over 16 bits by 4 multifunction data acquisition (DAQ) boards\n(National Instruments\\footnote{\\tiny{\\url{http:\/\/www.ni.com\/}}} M-6225-PCI),\nproviding 80 analog inputs each and synchronized to the same sample\nclock by a RTSI\\footnote{Real-Time System Integration, a bus used to share and exchange\ntiming and control signals between multiple boards.} bus . The maximum data\nsampling is 2500\\,Hz and the dynamic range can be selected over 5\npredefined ranges. The four boards provide also 24 digital\ninput\/output lines each, some of them used for the generation of the\nbias reference frequency and to monitor the digital reference signals\n(sync\/blank) of the wobbler. For the time synchronization of the data\nto the APEX control software \\citep[APECS,][]{2006A&A...454L..25M} the\ndata acquisition system is equipped with a precision time interface\n(PCI-SyncClock32\\footnote{\\tiny{\\url{http:\/\/www.brandywinecomm.com}}} from Brandywine Communications, Tustin, CA, USA)\nsynchronized to the station GPS clock via\nIRIG-B\\footnote{Inter Range Instrumentation Group, standardized time code format.} time code signal.\n\nThe AC bias reference frequency is provided by the data acquisition\nsystem as a submultiple of the sampling frequency, thus synchronizing\nthe bias to the data sampling. Typical values used for observations\nare 1\\,kHz for the sampling rate and 333\\,Hz for the AC bias. The\nbackend computer has two network adapters: one is connected to the\nlocal area network, the other one is exclusively used for the output\ndata stream and is connected in a private direct network with\nthe bridge computer (see also Sect.~\\ref{sec:bridge}).\n\nThe data acquisition software is entirely written using\nLabVIEW\\footnote{\\tiny{\\url{http:\/\/www.ni.com\/labview\/}}} (National Instruments).\nThe drivers for the data acquisition hardware are provided by the NI-DAQmx\\footnote{\\tiny{\\url{http:\/\/www.ni.com\/dataacquisition\/nidaqmx.htm}}} package.\nCustom drivers for LabVIEW have been developed to access the GPS clock interface.\nThe backend software runs a server to stream the output data to the bridge computer and allows remote control\nand monitoring of the operation via a CORBA\\footnote{Common Object Request Broker Architecture, a set of\nstandards which define the protocol for interaction between the objects of a distributed system} object interfaced\nto the APECS through the local area network.\n\\subsection{Anti-aliasing filtering and downsampling}\\label{sec:bridge}\nThe amplification units of LABOCA use the bias signal, which is a square waveform,\nas reference to operate real-time demodulation of the AC-biased bolometer signals.\nTherefore, all the frequencies present in the bolometer readout lines end up aliased\naround the odd-numbered harmonics of the bias frequency. Microphonics pickup by the\nhigh-impedance bolometers at a few resonant frequencies can produce a forest of lines in\nthe final readout, polluting even the lower part of the post-detection frequency band,\nwhere the astronomical signals are expected.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.96\\linewidth, bb=0 0 726 551,clip]{1454fig08light.eps}\n\\caption\n{\nNoise spectra for selected bolometers on a room temperature absorber (capped optics). \nThe signals, sampled at 1\\,kHz, were downsampled in real-time to 25\\,Hz by the bridge computer.\nThe spectra are free of microphonic pick-ups and show the $1\/f$ noise onset at $\\sim$0.1\\,Hz.\n}\n\\label{fig:noisespectra}\n\\end{figure}\nTo overcome this, we introduced an intermediate stage into the\nsampling scheme, the so-called {\\it bridge computer}. The bolometer\nsignals, acquired by the backend at a relatively high sampling rate\n(usually 1\\,kHz), well above the rolling-off of the anti-alias filters\nembedded in the amplifiers, are sent to the bridge computer where they\nare digitally low-pass filtered and then downsampled to a much lower\nrate (usually 25\\,Hz), more appropriate for the astronomical signals\nproduced at the typical scanning speeds (see also\nSect.~\\ref{sec:obsmodes}). The digital real-time anti-alias filtering\nand downsampling is performed by a non-recursive convolution filter\nwith a Nutall window such that its rejection at the Nyquist frequency\nis $\\sim$3\\,dB and falls steeply to $\\sim$100\\,dB soon beyond that.\nThe bias reference frequency was accordingly selected to maximize the\nastronomically useful post-detection bandwidth. The resulting\nbolometer signals are generally white between 0.1--12.5\\,Hz and free\nof unwanted microphonic interference (see Fig.~\\ref{fig:noisespectra}).\n\nTo seamlessly integrate the bridged readout into the APEX control\nsystem, the bridge computer also acts as a fully functional virtual\nbackend that forwards all communication between the actual backend\ncomputer and the control system, while intercepting and reinterpreting\nany commands of interest for the downsampling scheme.\n\\subsection{Frontend computer}\\label{sec:fepc}\nThe so-called {\\it frontend computer} communicates with the hardware\nof LABOCA through the local area network. It is devoted to monitor and\ncontrol most of the electronics of the instrument (e.g.~monitoring of\nall the temperature stages, control of the sorption cooler,\ncalibration unit, power lines\\dots) and also provides a CORBA object\nfor interfacing to APECS, allowing remote operation of the system.\nThe frontend software is entirely written using LabVIEW and custom\ndrivers have been developed for some hardware devices embedded in LABOCA.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.94\\linewidth, bb=33 3 259 420, clip]{1454fig09light.eps}\n\\caption{\nExamples of raster-spiral patterns. The two plots show the scanning pattern of the central beam\nof the array in horizontal coordinates. The compact pattern shown in\nthe top panel is optimized to map the field of view of LABOCA.\nThe large scale map shown in the bottom panel consists of 25 raster\npositions and covers a field of $0.5\\times0.5$ degrees. The complete\nscan required 19 minutes and the final map covers about\n$2000\\times2000$ arcseconds with uniform residual rms noise.\n}\n\\label{fig:spiralrast}\n\\end{figure}\n\n\\section{Observing modes}\\label{sec:obsmodes} \n\\subsection{Mapping modes}\\label{sec:mapmodes}\nIn order to reach the best signal-to-noise ratio using the fast scanning technique \\citep{2001A&A...379..735R}\nwith LABOCA, the frequencies of the signal produced by scanning across the source need to match the white noise part of\nthe post-detection frequency band (0.1--12.5\\,Hz, see Fig.~\\ref{fig:noisespectra}), mostly above the frequencies of the\natmospheric fluctuations. Thus, with the $\\sim$19$''$~beam (see Sect.~\\ref{sec:beam}),\nthe maximum practical telescope scanning speed for LABOCA is about $4'$\/s.\nThis is also the limiting value to guarantee the required accuracy in the telescope position information of each sample.\nThe minimum scanning speed required for a sufficient source modulation depends on the atmospheric stability and\non the source structure and is typically about $30''$\/s.\nThe APEX control system currently supports two basic scanning modes: on-the-fly maps (OTF) and spiral scanning patterns.\n\\subsubsection{Spirals}\nSpirals are done with a constant angular speed and an increasing radius, therefore the linear scanning velocity\nis not constant but increases with time. We have selected two spiral modes of 20\\,s and 35\\,s integration time,\nboth producing fully sampled maps of the whole FoV with scanning velocities limited between $1'$\/s and $2\\rlap{.}'5$\/s.\nThe spiral patterns are kept compact (maximum radius $\\lesssim2'$), the scanned area on the sky is only slightly\nlarger than the FoV and most of the integration time is spent on the central $11'$ of the array.\nThese spirals are the preferred observing modes for pointing scans on sources with flux densities down to a few Jy.\n\nFor fainter sources, the basic spiral pattern can be combined with a raster mapping mode (raster-spirals) on a grid\nof pointing positions resulting in an even denser sampling of the maps and longer integration time (see Fig.~\\ref{fig:spiralrast}, top panel).\nThese compact raster-spirals give excellent results for sources smaller than the FoV of LABOCA\nand are also suitable for integrations of very faint sources.\n\nThe flexibility in the choice of the spiral parameters also allows spiral\nobserving patterns to be used to map fields much larger than the FoV.\nThe bottom panel of Fig.~\\ref{fig:spiralrast} shows an example of\nraster of spirals optimized to give an homogeneous coverage across\na field of $0.5\\times0.5$ degrees. In this case, the spirals start\nwith a large radius and follows an almost circular scanning pattern\nfor each raster position. This mapping mode is very useful for\ncosmological deep field surveys since co-adding several such\nraster-spiral scans, taken at different times and thus at different\norientations, provides an optimal compromise between telescope\noverheads, uniform coverage and cross-linking of individual map\npositions \\citep[see ][]{Kovacs2008}.\n\n\\subsubsection{On-the-fly maps (OTF)}\nOTF scans are rectangular scanning patterns produced moving\nback-and-forth along alternating rows with a linear constant speed and\naccelerating only at the turnarounds. They can be performed in\nhorizontal or equatorial coordinates and the scanning direction can be\nrotated relatively to the base system for both coordinate systems. OTF\npatterns have been tested for maps on the scales of the FoV up to long\nslews across the plane of the Milky Way (2 degrees). Small\ncross-linked OTFs (of size $\\sim$FoV of LABOCA) give results\ncomparable to the raster-spirals \\citep{Kovacs2008}, but the\noverheads are much larger at a scanning speed of $2'$\/s. For larger\nOTFs the relative overheads decrease.\n\\subsection{Ancillary modes}\n\\subsubsection{Point}\nThe standard pointing procedure consists of one subscan in spiral\nobserving mode and results in a fully sampled map of the FoV of\nLABOCA. The pointing offsets relative to the pointing model are\ncomputed via a two-dimensional Gaussian fit to the source position in\nthe map using a BoA pipeline script (see Sect.~\\ref{sec:redu}). Note\nthat this pointing procedure is not limited to pointing scans of the\ncentral channel of the array but works independently of the reference\nbolometer, thus allowing pointing scans centered on the most sensitive\npart of the array.\n\\subsubsection{Focus}\nThe default focusing procedure is made of 10 subscans at 5 different\nsubreflector positions and 5 seconds of integration time each. This\nis the only observing mode without scanning telescope motion.\nAs a result, we are currently restricted to sources brighter than\nthe atmospheric variations (Mars, Venus, Saturn and Jupiter).\nHowever, initial tests confirmed that using the wobbler to modulate the\nsource signal allows focusing on weaker sources, too.\nThis is the only observing mode, so far, for which the use of the wobbler with LABOCA has been tested.\n\\subsubsection{Skydips}\\label{sec:skydips}\nThe attenuation of the astronomical signals due to the atmospheric\nopacity is determined with skydips. These scans measure the power of\nthe atmospheric emission as a function of the airmass while tipping\nthe telescope from high to low elevation. A skydip procedure consists\nof two steps: a hot-sky calibration scan, to provide an absolute\nmeasurement of the sky temperature, followed by a continuous tip scan\nin elevation; see also Sect.~\\ref{sec:opacity}.\n\\section{Performance on the sky and sensitivity}\\label{sec:performance}\n\\subsection{Number of channels}\\label{sec:numchans}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.98\\linewidth, bb=65 65 535 525, clip, angle=-90]{1454fig10light.eps}\n\\caption{\nFootprint of LABOCA on sky, measured with a beam map on the planet Mars.\nThe ellipses represent the FWHM shape of each beam on sky, as given by a\ntwo-dimensional Gaussian fit to the single-channel map of each\nbolometer. Only bolometers with useful signal-to-noise ratios are\nshown in this map. See also Fig.~\\ref{fig:sensitivity}.\n}\n\\label{fig:beams}\n\\end{figure}\nAt the time of the commissioning, the number of channels with sky\nresponse was 266 (90\\% of the nominal 295 bolometers). Of these, 6\nchannels show cross-talk and 12 channels have low sensitivity (less\nthan 10\\% of the mean responsivity). Two additional bolometers have\nbeen blinded by blocking their horn antennas with absorber material so\nthey can be used to monitor the temperature fluctuations of the array.\nThe number of channels used for astronomical observations is therefore\n248 (84\\% yield, see Fig.~\\ref{fig:beams}).\n\\subsection{Array parameters}\nPosition on sky and relative gain of each bolometer are derived from\nfully sampled maps (hereafter called {\\it beam maps}) of the planets\nMars and Saturn (see Fig.~\\ref{fig:beams}), besides giving a realistic\npicture of the optical distortions over the FoV. Variations among\nmaps were found to be within a few arc seconds for the positions and\nbelow 10\\% for peak flux densities. A table with average receiver\nparameters (RCP)\\footnote{The latest RCP table is available\nat:\\\\\n\\tiny{\\url{http:\/\/www.apex-telescope.org\/bolometer\/laboca\/calibration\/LABOCA-centred.rcp}}}\nis periodically computed from beam maps and implemented in the BoA software (see Sect.~\\ref{sec:redu}).\nThe accuracy of the relative bolometer positions from this master RCP is\ntypically below $1''$ (5\\% of the beam size) and the gain accuracy is\nbetter than 5\\%, confirming the good quality (small distortion over\nthe entire FoV) of the tertiary optics (see also\nSect.~\\ref{sec:optics}).\n\\subsection{Beam shape}\\label{sec:beam}\n\\begin{figure}[t] \n\\centering\n\\includegraphics[width=.98\\linewidth, bb=0 0 486 453, clip]{1454fig11light.eps} \n\\caption\n{\nRadial profile of the LABOCA beam derived by averaging the beams of all 248 functional bolometers\nfrom fully sampled maps on Mars. The error bars show the standard deviation.\nThe main beam is well described by a Gaussian with a FWHM of $19\\rlap{.}''2\\pm0\\rlap{.}''7$\nand starts to deviate at $-$20\\,dB.} \n\\label{fig:beamshape}\n\\end{figure}\nThe LABOCA beam shape was derived for individual bolometers from fully\nsampled maps on Mars (see Fig.~\\ref{fig:beams}) as well as on pointing\nscans on Uranus and Neptune. Both methods lead to comparable results\nand give an almost circular Gaussian with a FWHM of\n$19\\farcs2\\pm0\\farcs7$ (after deconvolution from the source\nand pixel sizes). We also investigated the error beam pattern of LABOCA\non beam maps on Mars and Saturn (see Fig.~\\ref{fig:beamshape}). The\nbeam starts to deviate from a Gaussian at $-$20~dB (1\\% of the peak\nintensity). The first error beam pattern can be approximated by a\nGaussian with a peak of $-18.3$~dB and a FWHM of~$70''\\pm5''$, the\nsupport legs of the subreflector are visible at the $-25$~dB (0.3\\%\nlevel). The fraction of the power in the first error beam is $\\sim$18\\%.\n\\subsection{Calibration}\\label{sec:calibra}\nThe astronomical calibration was achieved on Mars, Neptune and Uranus\nand a constant conversion factor of $6.3\\pm0.5$\\,Jy\\,beam$^{-1}$\\,\\microV$^{-1}$ was\ndetermined between LABOCA response and flux density. For the determination\nof the calibration factor we have used the fluxes of planets determined with\nthe software ASTRO\\footnote{GILDAS package, \\tiny{\\url{http:\/\/www.iram.fr\/IRAMFR\/GILDAS}}}. \nThe overall calibration accuracy for LABOCA is about 10\\%. \nWe have also defined a list of secondary calibrators\nand calibrated them against the planets (see Appendix~\\ref{sec:appendix}).\nIn order to improve the absolute flux determination, the calibrators\nare observed routinely every $\\sim$2~hours between observations of\nscientific targets.\n\\subsection{Sky opacity determination}\\label{sec:opacity}\nAtmospheric absorption in the passband of LABOCA attenuates the astronomical signals as $\\exp(-\\tau_{\\rm los})$\nwhere the line of sight optical depth $\\tau_{\\rm los}$ can be as high as 1 for observations at low elevation\nwith 2\\,mm of precipitable water vapor (PWV), typical limit for observations with LABOCA.\nThe accuracy of the absolute calibration, therefore, strongly depends on the precision in the determination of $\\tau_{\\rm los}$.\n\nWe use two independent methods for determining $\\tau_{\\rm los}$. \nThe first one relies on the PWV level measured every minute by the APEX radiometer broadly along the line of sight.\nThe PWV is converted into $\\tau_{\\rm los}$ using an atmospheric transmission model \\citep[ATM,][]{2001ITAP...49.1683P} and\nthe passband of LABOCA (see Sect.~\\ref{sec:passband}). The accuracy of this approach is limited by the knowledge of the passband,\nthe applicability of the ATM and the accuracy of the radiometer.\n\nThe second method uses skydips (see Sect.~\\ref{sec:skydips}).\nAs the telescope moves from high to low elevation, $\\tau_{\\rm los}$ increases with airmass.\nThe increasing atmospheric load produces an increasing total power signal converted\nto effective sky temperature ($T_{\\rm eff}$) by direct comparison with a reference hot load.\nThe dependence of $T_{\\rm eff}$ on elevation is then fitted to determine the zenith opacity~$\\tau$,\nused as parameter \\citep[see, e.g.,][]{2004MNRAS.354..621C}.\n\nSince LABOCA is installed in the Cassegrain cabin of APEX, when performing skydips the receiver suffers a wide,\ncontinuous rotation (about 70 degrees in 20 seconds), which affects the stability of the sorption cooler,\nthus inducing small variations of the bolometers temperature ($\\sim$1--2\\,mK).\nThese temperature fluctuations mimic an additional total power signal with amplitude comparable to the atmospheric signal.\nThe bolometers temperature, however, is monitored with high accuracy by the~\\element[][3]{He}-stage thermometer (see Sect.~\\ref{sec:thermonitor})\nand by the two blind bolometers (see Sect.~\\ref{sec:numchans}), making possible a correction of the skydip data.\n\nThe values of $\\tau$ resulting from the skydip analysis are robust,\nyet up to $\\sim$30\\% higher than those obtained from the radiometer.\nThere are several possible explanation for the discrepancy:\nit could be the result of some incorrect assumptions going into the skydip model (e.g.~the sky temperature),\nor the model itself may be incomplete. The non-linearity of the bolometers can be another factor.\nThe detector responsivities are expected to change with the optical load as $\\sim\\exp(-\\gamma\\,\\tau_{\\rm los})$ (to first order in $\\tau$),\nwhere $\\gamma$ can be related to the bolometer constants \\citep{Mather1984} and the optical configuration.\nCombined with the sky response, the bolometer non-linearities would increase the effective skydip $\\tau$ by a factor $(1+\\gamma)$.\n\nOur practical approach to reconciling the results obtained with the two methods has been to use a linear combination\nof radiometer and skydip values such that it gives the most consistent calibrator fluxes\nat all elevations\\footnote{A text file (BoA readable, see Sect.~\\ref{sec:redu}) containing the zenith opacities\ncalculated from the skydips and the radiometer is available at:\\\\\n\\tiny{\\url{http:\/\/www.apex-telescope.org\/bolometer\/laboca\/calibration\/opacity\/}}}.\nThe excellent calibration accuracy of LABOCA (see Appendix~\\ref{sec:appendix}) underscores this approach.\n\\subsection{Sensitivity}\\label{sec:sensitivity}\nThe noise-weighted mean point-source sensitivity of the array\n(noise-equivalent flux density, NEFD) determined from on-sky\nintegrations, is 55\\,mJy\\,s$^{1\/2}$ (sensitivity per channel). This\nvalue is achieved only by filtering the low frequencies (hence large\nscale emission) to reject residual sky-noise. For extended emission,\nwithout low frequency filtering, there is a degradation of sensitivity\nto a mean array sensitivity of 80--100\\,mJy\\,s$^{1\/2}$ depending on\nsky stability. However, there are significant variations of the\nsensitivity across the array (see Fig.~\\ref{fig:sensitivity}, top).\n\\begin{figure}[t] \n\\centering \n\\includegraphics[width=.98\\linewidth, bb=0 -30 698 704, clip]{1454fig12toplight.eps}\n\\includegraphics[width=.98\\linewidth, bb= 108 57 348 296,clip]{1454fig12bottomlight.eps} \n\\caption{\n{\\it Top:} Effective point source sensitivities (after low frequencies filtering) of all the useful bolometers of LABOCA\n(color scale in Jy\\,beam$^{-1}$). The positions were determined from beam maps on Mars, thus providing a realistic picture of\nthe optical distortions over the FoV (see also Fig.~\\ref{fig:beams}).\n{\\it Bottom:} Distribution of the effective point source sensitivities on the LABOCA array (5\\,mJy\\,s$^{1\/2}$ binning).\nThe total mapping speed of the array is as if the 250 good bolometers were all identical at a level of\n54.5\\,mJy\\,beam$^{-1}$\\,s$^{1\/2}$ (thick black line). The median sensitivity is also shown (dotted line).\n}\n\\label{fig:sensitivity}\n\\end{figure}\n\nFor detection experiments of compact sources with known position,\nLABOCA can be centered on the most sensitive part of the array rather\nthan on the geometric center. This results in an improved point source\nsensitivity of $\\sim$40\\,mJy\\,s$^{1\/2}$ for compact mapping pattern\nlike spirals.\n\\subsection{Mapping speed and time estimate}\nThe relation between the expected residual rms map noise $\\sigma$, the surveyed\narea on the sky and the integration time can be expressed as\n\\begin{equation}\nt_{\\rm int} = \\frac{(X_{\\rm scan}+D)~(Y_{\\rm scan}+D)}{A_{\\rm beam}~N_{\\rm bol}} \\left(\\frac{ f_{\\rm samp}~NEFD~~e^{\\tau_{\\rm los}}}{\\sigma}\\right)^2~\\\\\n\\\\\n\\end{equation}\nwhere $X_{\\rm scan}$, $Y_{\\rm scan}$ are the dimensions of the area covered by\nthe scanning pattern, $D$ is the size of the array, $A_{\\rm beam}$ is the\narea of the LABOCA beam, $N_{\\rm bol}$ is the number of working\nbolometers, $t_{\\rm int}$ is the on source integration time, $NEFD$ is the\naverage array sensitivity, $f_{\\rm samp}$ is the number of grid points\nper beam and $\\tau_{\\rm los}$ the line-of-sight opacity.\nThis formula\\footnote{An online time estimator is available at:\\\\\n\\tiny{\\url{http:\/\/www.apex-telescope.org\/bolometer\/laboca\/obscalc\/}}}\ndoes not include sensitivity variations across the array and the increasing\nsparseness of data points toward the map edges (the latter effect\ndepends on the mapping mode). We have tested this estimate on deep\nintegrations for point like and extended sources and found a\nreasonable agreement in the measured rms noise of the processed maps.\n\\subsection{Noise behavior in deep integrations}\n\\begin{figure}[t] \n\\centering \n\\includegraphics[width=.90\\linewidth, bb=0 0 351 357, clip]{1454fig13light.eps} \n\\caption\n{\nNoise behavior for deep integrations has been studied by co-adding about 350 hours of\nblank-field data. The plot shows the effective residual rms noise on sky.\n}\n\\label{fig:noise} \n\\end{figure}\nIn order to study how the noise averages down with increasing\nintegration time, {about 350 hours of blank-field mapping observations}\nhave been co-added with randomly inverted signs to remove any sources while\nkeeping the noise structure. As shown in~Fig.~\\ref{fig:noise}, the\nnoise integrates down with t$^{-1\/2}$, as expected.\n\\section{Data reduction}\\label{sec:redu} \nLABOCA data are stored in MB-FITS format (Multi-Beam FITS) by the data\nwriter embedded in APECS. A new software package has been\nspecifically developed to reduce LABOCA data: the Bolometer array data\nAnalysis software (BoA). It is mostly written in the Python language,\nexcept for the most computing demanding tasks, which are written in\nFortran90.\n\nBoA was first installed and integrated in APECS in early 2006. An extensive\ndescription of its functionalities will be given in a separate paper (Schuller\net al.~in prep.). In this Section, we outline the most important features\nfor processing LABOCA data.\n\\subsection{On-line data reduction}\nDuring the observations carried out at APEX, the on-line data\ncalibrator (as part of APECS) performs a quick data reduction of each\nscan to provide the observer with a quick preview of the maps or\nspectra being observed. This is of particular importance for the\nbasic pointing and focus observing modes, for which the on-line\ncalibrator computes and sends back to the observer the pointing\noffsets or focus corrections to be applied. For both focus and\npointing scans, only a quick estimate of the correlated noise (see\nbelow) is computed and subtracted from the data. The focus correction\nis derived from a parabolic fit to the peak flux measured by the\nreference bolometer as a function of the subreflector position. For\npointing scans, the signals of all usable channels are combined into a\nmap of the central $5'\\times 5'$ area, in horizontal coordinates. A\ntwo-dimensional elliptical Gaussian is then fitted to the source in\nthis map, which gives the pointing offsets, as well as the peak flux\nand the dimensions of minor and major axis of the source.\n\\subsection{Off-line data reduction}\nThe BoA software can also be used off-line to process any kind of\nbolometer data acquired at APEX. The off-line BoA runs in the\ninteractive environment of the Python language. In a typical off-line\ndata reduction session, several scans can be combined together, for\ninstance to improve the noise level on deep integrations, or to do\nmosaicing of maps covering adjacent areas. The result of any data\nreduction can be stored in a FITS file using standard world coordinate\nsystem (WCS) keywords, which can then be read by other softwares for\nfurther processing (e.g.~source extraction, or overlay with ancillary\ndata).\n\nThe common steps involved in the processing of LABOCA data are the\nfollowing: \\begin{itemize} \\item Flux calibration. A correct scaling\nof the flux involves, at least, two steps: the opacity correction and\nthe counts-to-Jy conversion. The zenith opacity is derived from\nskydip measurements (see Sect.~\\ref{sec:skydips} and\nSect.~\\ref{sec:opacity}), and the line of sight opacity also depends\non the elevation. The counts-to-Jy factor has been determined during\nthe commissioning of the instrument, but additional correction factors\nmay be applied, depending on the flux measured on calibrators with\nknown fluxes (see Sect.~\\ref{sec:calibra}). \\item Flagging of bad\nchannels. Bolometers not responding or with strong excess noise are\nautomatically identified from their rms noise being well outside the\nmain distribution of the rms noise values across the array. They can\nbe flagged, which means that the signal that they recorded is not used\nany further in the processing. \n\\begin{figure}[t] \n\\centering\n\\includegraphics[width=.96\\linewidth, bb=0 0 340 244,clip]{1454fig14toplight.eps} \n\\includegraphics[width=.96\\linewidth, bb=0 0 339 245, clip]{1454fig14bottomlight.eps} \n\\caption\n{ \nTime series of a bolometer signal on blank sky at consecutive steps during the data reduction process with BoA. The signals, labeled from 1 to\n5, are shown for:\n1) original signal on sky;\n2) after correcting for system temperature variations using the blind bolometers ($\\sim$200\\,\\microK during this scan);\n3) after median skynoise removal over the full array, one single iteration;\n4) after median skynoise removal over the full array, 10 iterations;\n5) after correlated noise removal, grouping the channels by amplifier and cable, 5 iterations (shifted to a $-$3 level, for clarity).\n}\n\\label{fig:reduction} \n\\end{figure} \n\\item Flagging of stationary\npoints. The data acquired when the telescope was too slow to produce\na signal inside the useful part of the post-detection frequency band\nof LABOCA (e.g.~below 0.1\\,Hz, see Sec.~\\ref{sec:mapmodes} and\nFig.~\\ref{fig:noisespectra}) can be flagged. Data obtained when the\ntelescope acceleration is very high may show some excess noise, and\ncan also be flagged. \n\\item Correlated noise removal. This can be\ndone using a Principal Components Analysis (PCA), or a median noise\nremoval method. In the latter case, the median value of all\n(normalized) signals is computed at each timestamp, and subtracted\nfrom the signal of each channel (with appropriate relative gains).\nThis can be performed using all beams at once or better on groups of\nselected beams. In fact, some groups of channels sharing the same\nelectronics subsystem (e.g.~amplifier box, flat cable) can show strong correlation\nand removing the median signal on those groups of channels greatly improves\ntheir signal-to-noise ratio. However, it should be noted that an astronomical source\nwith extended uniform emission on scales of several arcmin or larger would mimic the\ncorrelated noise behavior of groups of channels. Therefore,\nsubtracting the median noise results in filtering out some fraction of\nthe extended emission, depending on the size and morphology of the source. \n\\item Despiking. Data points that deviate by more than a\ngiven factor times the standard deviation in each channel can be flagged. \n\\item Data weighting and map making. To build a map in\nhorizontal or equatorial coordinates, the data of all usable channels\nare projected onto a regular grid and co-added, using a weighted\naverage, with weights computed as 1\/rms$_{\\rm c}^2$, where rms$_{\\rm c}$\nrefers to the rms noise of an individual channel. The channel rms\nnoise can either be the standard deviation of the data over the full\ntime line, or it can be computed on a sliding window containing a\ngiven number of integrations. \\end{itemize} A visual example of\ncorrelated noise removal with BoA is given\nin~Fig.~\\ref{fig:reduction}.\n\nAdditional (optional) steps that can improve the final reduction\ninclude: removing slow variations from the signal, by subtracting a\npolynomial baseline or by filtering out low frequencies in the Fourier\ndomain; smoothing of the map with a Gaussian kernel of a given size.\n\nThe map resulting from a full reduction can be used as a model of the\nastronomical source to mask the data, in order to repeat some\ncomputations without using data points corresponding to the source in\nthe map. This iterative scheme helps in the difficult task to recover\nthe extended emission and reduces the generation of artifacts around\nstrong sources. In the presence of bright sources in the map, a\ntypical data reduction should first perform the reduction steps as\nlisted above with conservative parameters (for example, using high\nenough thresholds in the despiking to avoid flagging out the strong\nsources), and should then read again the raw data, use the resulting\nmap to temporarily flag out datapoints corresponding to bright\nsources, and repeat the full process with less conservative\nparameters.\n\nIn addition to the data processing itself, BoA allows visualization of\nthe data in different ways (signal vs. time, correlation between channels,\npower spectra or datagrams), as well as the telescope pattern, speed\nand acceleration. Finally, BoA also includes a simulation module,\nwhich can be used to investigate the mapping coverage for on-the-fly\nmaps, spirals and raster of spirals, given the bolometer array\nparameters.\n\\section{Science with LABOCA}\\label{sec:science}\nBecause of its spectral passband, centered at the wavelength of\n870\\,\\micron (see Fig.~\\ref{fig:passband}), LABOCA is particularly\nsensitive to thermal emission from cold objects in the Universe which\nis of great interest for a number of astrophysical research fields.\n\\subsection{Planet Formation}\nThe study of Kuiper Belt Objects in the Solar System as well as\nobservations of debris disks of cold dust around nearby main sequence\nstars can give vital clues to the formation of our own planetary\nsystem and planets in general. With its angular resolution of~19\\arcsec\n(see Fig.~\\ref{fig:beamshape}) LABOCA can resolve the debris disks of\nsome nearby stars.\n\\subsection{Star Formation in the Milky Way}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.99\\linewidth,bb=20 20 432 786, clip]{1454fig15light.eps} \n\\caption\n{\nA map of the galactic center, extracted from the project ATLASGAL (F. Schuller et al., in prep.).\nThe map is 2$\\times$4 degrees large, the residual noise is 50\\,mJy\\,beam $^{-1}$ and it required only 6~hours of\nobserving time. Data reduced with the BoA package.\n}\n\\label{fig:atlasgal}\n\\end{figure}\nThe outstanding power of LABOCA in mapping large areas of sky with\nhigh sensitivity (see Fig.~\\ref{fig:atlasgal}) makes it possible,\nfor the first time, to perform unbiased surveys of the distribution\nof the cold dust in the Milky Way.\nAs the dust emission at 870\\,\\micron is typically optically thin, it is\na direct tracer of the gas column density and gas mass. Large scale\nsurveys in the Milky Way will reveal the distribution and gas\nproperties of a large number of pre-star cluster clumps and\npre-stellar cores in different environments and evolutionary\nstates. Equally importantly, they provide information on the structure\nof the interstellar medium on large scales at high spatial resolution,\nlittle explored so far. Such surveys are vital to improve our\nunderstanding of the processes that govern star formation as well as\nthe relation between the clump mass spectrum and the stellar initial\nmass function. Large unbiased surveys are also critical for finding\nprecursors of high-mass stars, which are undetectable at other\nwavelengths due to the high obscuration of the massive cores they are\nembedded in. LABOCA will help to obtain a detailed understanding of\ntheir evolution. In addition, deep surveys of nearby, star-forming\nclouds, will allow study of the pre-stellar mass function down to the\nbrown dwarf regime.\n\\subsection{Cold gas in Galaxies}\nThe only reliable way to trace the bulk of dust in galaxies is through\nimaging at submillimeter wavelengths. It is becoming clear that most\nof the dust mass in spiral galaxies lies in cold, low\nsurface-brightness disks, often extending far from the galactic\nnucleus, as in the case of the starburst galaxy \\object{NGC 253} or of \\object{NGC 5128}\n\\citep[\\object{Cen A}, see Fig.~\\ref{fig:ngc5128} and][]{2008A&A...490...77W}.\n\\begin{figure}[t] \n\\centering\n\\includegraphics[width=.99\\linewidth, bb=0 0 416 307, clip]{1454fig16light.eps} \n\\caption\n{\nMap of the nearby galaxy Centaurus A \\citep[\\object{NGC 5128}, from ][]{2008A&A...490...77W}.\nThe central dust disk has a pronounced S-shape. Our LABOCA image shows, for the first time at\nsubmillimeter wavelengths, the synchrotron emission from the radio jets and\nthe inner radio lobes. Its distribution follows closely the known\nradio emission at cm wavelengths. The color scale is in units\nof Jy\\,beam$^{-1}$. The residual rms noise is 4\\,mJy\\,beam$^{-1}$ and\nthe total integration time is 5~hours.\n}\n\\label{fig:ngc5128}\n\\end{figure} \nUnderstanding this component is critically important as\nit dominates the total gas mass in galaxies and studies of the Schmidt\nlaw based on \\ion{H}{i} observations only can heavily underestimate\nthe gas surface density in the outer parts.\n\nIn addition to studying individual nearby galaxies, LABOCA will be\nvital for determining the low-$z$ benchmarks, such as the local\nluminosity and dust mass functions, which are required to interpret\ninformation from deep cosmological surveys.\n\\subsection{Galaxy formation at high redshift}\nDue to the negative-K correction, submillimeter observations offer\nequal sensitivity to dusty star-forming galaxies over a redshift range\nfrom $z\\sim$1--10 and therefore provide information on the star\nformation history at epochs from about half to only 5\\% of the present\nage of the universe. Recent studies have shown that the volume density\nof luminous {\\it submillimeter galaxies} (SMGs) increases over a\nthousand-fold out to $z\\sim$2 \\citep{2005ApJ...622..772C}, and thus,\nin contrast to the local Universe, luminous obscured galaxies at high\nredshift could dominate the total bolometric emission from all\ngalaxies at early epochs. The mass-tracing property of submillimeter\ndust emission (see Sect.~\\ref{sec:intro}) allows us to make a direct estimate\nof the star formation rates (SFRs) of these objects. The generally high\nobserved SFRs suggest that approximately half of all the stars that\nhave formed by the present day may have formed in highly obscured\nsystems which remain undetected at optical or near-infrared\nwavelengths. One example for such system is the submillimeter galaxy\n\\object{SMM J14009+0252} (see Fig.~\\ref{fig:smm14009}), which is strong in the\nsubmillimeter range, has a 1.4\\,GHz radio counterpart, but no obvious\ncounterpart in deep K-band images \\citep{2000MNRAS.315..209I}.\nClearly it is critical to include these highly obscured sources in\nmodels of galaxy formation to obtain a complete understanding of the\nevolution of galaxies.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.99\\linewidth, bb=28 192 567 610, clip]{1454fig17light.eps}\n\\caption\n{\nLABOCA images of \\object{SMM J14011+0252} (left) and \\object{SMM J14009+0252} (right) smoothed to $25''$ resolution.\nSMM\\,J14011 is at a redshift of $z=2.56$ while SMM\\,J14009 has no reliable redshift determination.\nThe noise level of the map is about 2.5\\,mJy\\,beam$^{-1}$.\n}\n\\label{fig:smm14009}\n\\end{figure}\nWith its fast mapping capabilities LABOCA allows us to map fields of a\nhalf square degree, a typical size of a deep cosmological field\nsurveyed at other wavelengths, down to the confusion limit in a\nreasonable amount of observing time. This will also greatly improve\nthe statistics of high redshift galaxies detected at submillimeter\nwavelengths. See \\citet{Beelen2008} for the first deep LABOCA\ncosmology imaging.\n\\section{Future plans}\\label{sec:future}\nIn collaboration with the Institute for Photonics Technology (IPHT, Jena, Germany)\nwe are working on a new bolometer array, LABOCA-2, using\nsuperconducting technology. The new array will use superconducting\nthermistors (transition edge sensor, TES), planar dipole absorbers,\nSQUID (Superconducting Quantum Interference Device) multiplexing and\namplification. A system already using the same technology, called\nSubmillimeter APEX Bolometer Camera (SABOCA) is going to be\ncommissioned as facility instrument on the APEX telescope for\noperation in the 350\\,\\micron atmospheric window. Additionally, given\nthe low sensitivity of superconducting bolometers to microphonics, it\nwill be possible to move from the wet cryostat to a pulse-tube cooler,\nas already successfully tested at MPIfR in Bonn, improving\nconsiderably the conditions of operability of the system.\n\\begin{acknowledgements}\nThe authors would like to thank the staff at the APEX telescope for their\nsupport during installation and commissioning of the instrument. Special\nthanks to Lars-{\\AA}ke Nyman (ESO, APEX station manager at the time of the\ncommissioning) for his invaluable help and for his belief in the project\neven in the most difficult moments.\nE.K. enjoyed the hospitality and help provided by the staff and many student members\nof the Microfabrication Facility of UC Berkeley during the manufacture of the LABOCA wafer.\n\\end{acknowledgements} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}