Integral Formulas and asymptotic behavior of lattice points in complex hyperbolic space
This paper deals with the \(\Gamma\)-lattice points problem associated to a discrete subgroup of motions \(\Gamma\) in the complex hyperbolic space \(\mathbb{C} H^n\). We give two integral formulas for the local average of the number \(N(T, z, z')\) of \(\Gamma\)- lattice points in a sphere of...
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description | This paper deals with the \(\Gamma\)-lattice points problem associated to a discrete subgroup of motions \(\Gamma\) in the complex hyperbolic space \(\mathbb{C} H^n\). We give two integral formulas for the local average of the number \(N(T, z, z')\) of \(\Gamma\)- lattice points in a sphere of radius \(T\) in \(\mathbb{C} H^n\). The first on is in terms of the solution of the \(\Gamma\)-automorphic wave equation on \(\mathbb{C} H^n\) and the second is given in terms of the spectral function of the Laplace-Beltrami operator under \(\Gamma\)-automorphic boundary conditions. We use the obtained integral formulas to obtain an asymptotic behavior of the number \(N(T, z, z')\) as \(T\rightarrow \infty\), with an estimate of the remainder term. Our principal tools are the explicit solution of the wave equation on the complex hyperbolic space, special functions and spectral theory of the Laplace-Beltrami operator under \(\Gamma\) automorphic boundary conditions. |
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We give two integral formulas for the local average of the number \(N(T, z, z')\) of \(\Gamma\)- lattice points in a sphere of radius \(T\) in \(\mathbb{C} H^n\). The first on is in terms of the solution of the \(\Gamma\)-automorphic wave equation on \(\mathbb{C} H^n\) and the second is given in terms of the spectral function of the Laplace-Beltrami operator under \(\Gamma\)-automorphic boundary conditions. We use the obtained integral formulas to obtain an asymptotic behavior of the number \(N(T, z, z')\) as \(T\rightarrow \infty\), with an estimate of the remainder term. Our principal tools are the explicit solution of the wave equation on the complex hyperbolic space, special functions and spectral theory of the Laplace-Beltrami operator under \(\Gamma\) automorphic boundary conditions.</description><identifier>EISSN: 2331-8422</identifier><language>eng</language><publisher>Ithaca: Cornell University Library, arXiv.org</publisher><subject>Asymptotic properties ; Boundary conditions ; Integrals ; Operators (mathematics) ; Spectral theory ; Subgroups ; Wave equations</subject><ispartof>arXiv.org, 2020-04</ispartof><rights>2020. This work is published under http://arxiv.org/licenses/nonexclusive-distrib/1.0/ (the “License”). 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The first on is in terms of the solution of the \(\Gamma\)-automorphic wave equation on \(\mathbb{C} H^n\) and the second is given in terms of the spectral function of the Laplace-Beltrami operator under \(\Gamma\)-automorphic boundary conditions. We use the obtained integral formulas to obtain an asymptotic behavior of the number \(N(T, z, z')\) as \(T\rightarrow \infty\), with an estimate of the remainder term. 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subjects | Asymptotic properties Boundary conditions Integrals Operators (mathematics) Spectral theory Subgroups Wave equations |
title | Integral Formulas and asymptotic behavior of lattice points in complex hyperbolic space |
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