On the P\'olya conjecture for circular sectors and for balls
@inproceedings{Filonov2022OnTP,
title={On the P\'olya conjecture for circular sectors and for balls},
author={Nikolai Filonov},
year={2022}
}In 1954, G. P´olya conjectured that the counting function N (Ω , Λ) of the eigenvalues of the Laplace operator of the Dirichlet (resp. Neumann) boundary value problem in a bounded set Ω ⊂ R d is lesser (resp. greater) than (2 π ) − d ω d | Ω | Λ d/ 2 . Here Λ is the spectral parameter, and ω d is the volume of the unit ball. We prove this conjecture for both Dirichlet and Neumann boundary problems for any circular sector, and for the Dirichlet problem for a ball of arbitrary dimension. We…
One Citation
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References
SHOWING 1-10 OF 16 REFERENCES
P\'olya's conjecture for the disk: a computer-assisted proof
- Mathematics
- 2022
The celebrated Pólya’s conjecture (1954) in spectral geometry states that the eigenvalue counting functions of the Dirichlet and Neumann Laplacian on a bounded Euclidean domain can be estimated from…
Dirichlet and Neumann Eigenvalue Problems on Domains in Euclidean Spaces
- Mathematics
- 1997
Abstract We obtain here some inequalities for the eigenvalues of Dirichlet and Neumann value problems for general classes of operators (or system of operators) acting in L 2 ( Ω ) (or L 2 ( Ω , C m…
Bounds for zeros of some special functions
- Mathematics
- 1970
For n> 1 let bn and c. be zeros (ordered by in- creasing values) of u(x) and v(x), respectively, which are non- trivial solutions of u"+p(x)u=O and v"+q(x)v=O with contin- uous p(x) and q(x). It is…
On the remainder in the Weyl formula for the Euclidean disk
- Mathematics
- 2010
We prove a 2-terms Weyl formula for the counting function N(mu) of the spectrum of the Laplace operator in the Euclidean disk with a sharp remainder estimate O(mu^2/3).
On the Schrödinger equation and the eigenvalue problem
- Mathematics
- 1983
AbstractIf λk is thekth eigenvalue for the Dirichlet boundary problem on a bounded domain in ℝn, H. Weyl's asymptotic formula asserts that
$$\lambda _k \sim C_n \left( {\frac{k}{{V(D)}}}…
Bessel phase functions: calculation and application
- MathematicsNumerische Mathematik
- 2017
Some new properties of the modulus and phase functions and some asymptotic expansions derived from differential equation theory are given and a bound on the error of the first term of this asymPTotic expansion is found.