Sharp spectral estimates in domains of infinite volume

TL;DR

This paper develops a method to obtain sharp spectral estimates for the Dirichlet Laplace operator in infinite-volume domains, overcoming the limitations of phase-space volume-based estimates, with applications to specific geometries.

Contribution

It introduces a novel approach to derive uniform eigenvalue bounds in infinite-volume domains, extending previous semiclassical estimates beyond phase-space volume constraints.

Findings

Established sharp eigenvalue bounds in horn-shaped regions.

Extended results to Schrödinger operators on quasi-bounded domains.

Provided examples demonstrating the method's effectiveness.

Abstract

We consider the Dirichlet Laplace operator on open, quasi-bounded domains of infinite volume. For such domains semiclassical spectral estimates based on the phase-space volume - and therefore on the volume of the domain - must fail. Here we present a method how one can nevertheless prove uniform bounds on eigenvalues and eigenvalue means which are sharp in the semiclassical limit. We give examples in horn-shaped regions and so-called spiny urchins. Some results are extended to Schr\"odinger operators defined on quasi-bounded domains with Dirichlet boundary conditions.