### 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 language | English |
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Pages (from-to) | 473-482 |

Number of pages | 10 |

Journal | Russian Journal of Mathematical Physics |

Volume | 22 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Oct 2015 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Russian Journal of Mathematical Physics*,

*22*(4), 473-482. https://doi.org/10.1134/S106192081504007X

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

Research output: Contribution to journal › Article

*Russian Journal of Mathematical Physics*, vol. 22, no. 4, pp. 473-482. https://doi.org/10.1134/S106192081504007X

}

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 -