Families of non-tiling domains satisfying Pólya's conjecture

12/04/2023

We show the existence of classes of non-tiling domains satisfying Pólya’s conjecture in any dimension, in both the Euclidean and non-Euclidean cases. This is a consequence of a more general observation asserting that if a domain satisfies Pólya’s conjecture eventually, that is, for a sufficiently large order of the eigenvalues, and may be partioned into p nonoverlapping isometric sub-domains, with p arbitrarily large, then there exists an order p0 such that for p larger than p0 all such sub-domains satisfy Pólya’s conjecture. In particular, this allows us to show that families of sectors of domains of revolution with analytic boundary, and thin cylinders satisfy Pólya’s conjecture, for instance. We also improve upon the Li–Yau constant for general cylinders in the Dirichlet case.

Authors in the community:

• 

  P

  Pedro Simões Cristina de Freitas

  ist12267

  Affiliations and sources

  IST > DM

  Departamento de Matemática

  In: orcid

  ID: 148018468

• 

  I

  Isabel Maria da Costa Salavessa

  ist23588

  Affiliations and sources

  IST > GFM

  Grupo de Física Matemática

VoR - Version of Record

Journal of Mathematical Physics , AIP Publishing

https://pubs.aip.org/aip/jmp/article/64/12/121503/2925675/Families-of-non-tiling-domains-satisfying-Polya-s

Journal of Mathematical Physics

1

7

64

12

10.1063/5.0161050

mathematics - Mathematics

Keywords

• Laplace operator
• eigenvalues
• Pólya's conjecture

eng - English