Polya's conjecture in the presence of a constant magnetic field

TL;DR

This paper investigates the effects of a constant magnetic field on the eigenvalue estimates of the Dirichlet Laplacian, demonstrating that Polya's conjecture does not hold under these conditions.

Contribution

It provides the first sharp constants for semi-classical eigenvalue estimates of the magnetic Dirichlet Laplacian and disproves Polya's conjecture with a magnetic field.

Findings

Polya's conjecture is false with a magnetic field

Sharp constants for eigenvalue estimates are established

Magnetic field influences eigenvalue bounds significantly

Abstract

We consider the Dirichlet Laplacian with a constant magnetic field in a two-dimensional domain of finite measure. We determine the sharp constants in semi-classical eigenvalue estimates and show, in particular, that Polya's conjecture is not true in the presence of a magnetic field.