Distribution of the error term in the Weyl asymptotics for the Laplace operator on a two-dimensional torus and related lattice problems
type: Item
instance of: scholarly article
title: Distribution of the error term in the Weyl asymptotics for the Laplace operator on a two-dimensional torus and related lattice problems
zbMATH Open document ID: 0803.11052
DOI: 10.1215/S0012-7094-93-07015-9
author: Pavel M. Bleher
published in: Duke Mathematical Journal
publication date: 1995-01-03T00:00:00Z
review text: Let \(E_ n\) be the eigenvalues of the Laplace-Beltrami operator on a \(d\)-dimensional compact smooth closed Riemannian manifold \(M\), \(-\Delta \varphi_ n= E_ n \varphi_ n\) and let \(N(E)= \#\{ E_ n\leq E\}\) be the spectral function of this operator. A classical problem is: What is the asymptotic behavior of the error term (in the Weyl formula) \[ D(E)= N(E)- {{\text{Vol } M\cdot \text{Vol } \Omega_ d} \over {(2\pi)^ d}} E^{d/2}, \] where \(E\to\infty\), where \(\Omega_ d= \{x\in \mathbb{R}^ d\), \(| x|\leq 1\}\). In the present work the authors study this problem in the case when \(M\) is a two-dimensional torus \(\mathbb{R}^ 2/ (2\pi a_ 1 \mathbb{Z}\oplus 2\pi a_ 2\mathbb{Z})\) and obtain interesting results, in particular if \({{a_ 1} \over {a_ 2}}\) is transcendental. The results are extended to a general class of \((C^ 7\)-smooth convex) ovals containing the origin with positive curvature such that the Fourier frequencies of a certain normalized error term satisfy some linear independence properties.
Mathematics Subject Classification ID: 11P21
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 558979
zbMATH Keywords: eigenvalues
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: spectral function
zbMATH Keywords: asymptotic behavior of the error term
zbMATH Keywords: two-dimensional torus
reviewed by: Harald Rindler
MaRDI profile type: Publication
cites work: On the distribution of the number of lattice points inside a family of convex ovals
cites work: Distribution of the error term for the number of lattice points inside a shifted circle
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cites work: The distribution and moments of the error term in the Dirichlet divisor problem
cites work: ON THE NUMBER OF LATTICE POINTS INSIDE A RANDOM OVAL
full work available at URL: s0012-7094-93-07015-9
OpenAlex ID: W1500300291
description: scientific article; zbMATH DE number 558979