Eigenvalue inequalities for Schr\"odinger operators on unbounded Lipschitz domains

TL;DR

This paper establishes strict inequalities between eigenvalues of Schrödinger operators on unbounded Lipschitz domains under various boundary conditions, advancing understanding of spectral properties in such settings.

Contribution

It introduces new strict eigenvalue inequalities for Schrödinger operators on unbounded Lipschitz domains, including comparisons between different boundary conditions and operators.

Findings

Proves strict inequalities between Dirichlet and Neumann eigenvalues.

Establishes inequalities between eigenvalues of different elliptic operators.

Extends spectral inequality results to unbounded Lipschitz domains.

Abstract

Given a Schr\"odinger differential expression on an exterior Lipschitz domain we prove strict inequalities between the eigenvalues of the corresponding selfadjoint operators subject to Dirichlet and Neumann or Dirichlet and mixed boundary conditions, respectively. Moreover, we prove a strict inequality between the eigenvalues of two different elliptic differential operators on the same domain with Dirichlet boundary conditions.