Inequalities for eigenvalues of Schr\"odinger operators with mixed boundary conditions

TL;DR

This paper establishes inequalities relating eigenvalues of Schrödinger operators with mixed boundary conditions on convex domains, providing bounds and comparisons for different boundary setups in multiple dimensions.

Contribution

It introduces new inequalities between eigenvalues for mixed boundary conditions and pure Dirichlet conditions on convex and polyhedral domains, extending previous spectral bounds.

Findings

Inequalities between lowest eigenvalues for different boundary conditions

Comparison of higher order mixed eigenvalues with Dirichlet eigenvalues

Results applicable to both planar and higher-dimensional domains

Abstract

We consider the eigenvalue problem for the Schr\"odinger operator on bounded, convex domains with mixed boundary conditions, where a Dirichlet boundary condition is imposed on a part of the boundary and a Neumann boundary condition on its complement. We prove inequalities between the lowest eigenvalues corresponding to two different choices of such boundary conditions on both planar and higher-dimensional domains. We also prove an inequality between higher order mixed eigenvalues and pure Dirichlet eigenvalues on multidimensional polyhedral domains.