On a problem in geometry of numbers arising in spectral theory

Yu A. Kordyukov, A. A. Yakovlev

Research output: Contribution to journalArticle

Abstract

We study the lattice point counting problem for a class of families of domains in a Euclidean space. This class consists of anisotropically expanding bounded domains that remain unchanged along some fixed linear subspace and expand in directions orthogonal to this subspace. We find the leading term in the asymptotics of the number of lattice points in such family of domains and prove remainder estimates in this asymptotics under various conditions on the lattice and the family of domains. As a consequence, we prove an asymptotic formula for the eigenvalue distribution function of the Laplace operator on a flat torus in adiabatic limit determined by a linear foliation with a nontrivial remainder estimate.

Original languageEnglish
Pages (from-to)473-482
Number of pages10
JournalRussian Journal of Mathematical Physics
Volume22
Issue number4
DOIs
Publication statusPublished - 1 Oct 2015
Externally publishedYes

Fingerprint

Geometry of numbers
spectral theory
Spectral Theory
Lattice Points
Remainder
geometry
Subspace
Eigenvalue Distribution
Counting Problems
Laplace Operator
Foliation
Asymptotic Formula
Laplace transformation
Estimate
Expand
Euclidean geometry
Euclidean space
Bounded Domain
Torus
Distribution Function

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

On a problem in geometry of numbers arising in spectral theory. / Kordyukov, Yu A.; Yakovlev, A. A.

In: Russian Journal of Mathematical Physics, Vol. 22, No. 4, 01.10.2015, p. 473-482.

Research output: Contribution to journalArticle

@article{dbf2ad196a914e00b120d71e8072597a,
title = "On a problem in geometry of numbers arising in spectral theory",
abstract = "We study the lattice point counting problem for a class of families of domains in a Euclidean space. This class consists of anisotropically expanding bounded domains that remain unchanged along some fixed linear subspace and expand in directions orthogonal to this subspace. We find the leading term in the asymptotics of the number of lattice points in such family of domains and prove remainder estimates in this asymptotics under various conditions on the lattice and the family of domains. As a consequence, we prove an asymptotic formula for the eigenvalue distribution function of the Laplace operator on a flat torus in adiabatic limit determined by a linear foliation with a nontrivial remainder estimate.",
author = "Kordyukov, {Yu A.} and Yakovlev, {A. A.}",
year = "2015",
month = "10",
day = "1",
doi = "10.1134/S106192081504007X",
language = "English",
volume = "22",
pages = "473--482",
journal = "Russian Journal of Mathematical Physics",
issn = "1061-9208",
publisher = "Maik Nauka-Interperiodica Publishing",
number = "4",

}

TY - JOUR

T1 - On a problem in geometry of numbers arising in spectral theory

AU - Kordyukov, Yu A.

AU - Yakovlev, A. A.

PY - 2015/10/1

Y1 - 2015/10/1

N2 - We study the lattice point counting problem for a class of families of domains in a Euclidean space. This class consists of anisotropically expanding bounded domains that remain unchanged along some fixed linear subspace and expand in directions orthogonal to this subspace. We find the leading term in the asymptotics of the number of lattice points in such family of domains and prove remainder estimates in this asymptotics under various conditions on the lattice and the family of domains. As a consequence, we prove an asymptotic formula for the eigenvalue distribution function of the Laplace operator on a flat torus in adiabatic limit determined by a linear foliation with a nontrivial remainder estimate.

AB - We study the lattice point counting problem for a class of families of domains in a Euclidean space. This class consists of anisotropically expanding bounded domains that remain unchanged along some fixed linear subspace and expand in directions orthogonal to this subspace. We find the leading term in the asymptotics of the number of lattice points in such family of domains and prove remainder estimates in this asymptotics under various conditions on the lattice and the family of domains. As a consequence, we prove an asymptotic formula for the eigenvalue distribution function of the Laplace operator on a flat torus in adiabatic limit determined by a linear foliation with a nontrivial remainder estimate.

UR - http://www.scopus.com/inward/record.url?scp=84949186943&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84949186943&partnerID=8YFLogxK

U2 - 10.1134/S106192081504007X

DO - 10.1134/S106192081504007X

M3 - Article

AN - SCOPUS:84949186943

VL - 22

SP - 473

EP - 482

JO - Russian Journal of Mathematical Physics

JF - Russian Journal of Mathematical Physics

SN - 1061-9208

IS - 4

ER -