Improved semiclassical eigenvalue estimates for the Laplacian and the Landau Hamiltonian

TL;DR

This paper improves classical eigenvalue inequalities for the Laplacian and Landau Hamiltonian by introducing a multiplicative factor depending on domain geometry and eigenvalue parameters, applicable in magnetic fields.

Contribution

It provides enhanced bounds for Riesz means of eigenvalues, incorporating geometric factors and extending results to magnetic field scenarios.

Findings

Improved eigenvalue inequalities with geometric dependence.

Extension of inequalities to magnetic field contexts.

Quantitative bounds involving domain inradius and eigenvalue cut-off.

Abstract

The Berezin--Li--Yau and the Kr\"oger inequalities show that Riesz means of order $\geq 1$ of the eigenvalues of the Laplacian on a domain $\Omega$ of finite measure are bounded in terms of their semiclassical limit expressions. We show that these inequalities can be improved by a multiplicative factor that depends only on the dimension and the product $\sqrt\Lambda |\Omega|^{1/d}$, where $\Lambda$ is the eigenvalue cut-off parameter in the definition of the Riesz mean. The same holds when $|\Omega|^{1/d}$ is replaced by a generalized inradius of $\Omega$. Finally, we show similar inequalities in two dimensions in the presence of a constant magnetic field.