On the spectral gap of higher-dimensional Schr\"odinger operators on large domains

TL;DR

This paper investigates how the spectral gap of higher-dimensional Schr"odinger operators changes as the domain size increases, revealing different asymptotic behaviors depending on the potential's properties.

Contribution

It provides a detailed analysis of the asymptotic behavior of spectral gaps in large domains for Schr"odinger operators, highlighting conditions affecting whether the gap resembles that of the free Laplacian.

Findings

Spectral gap can asymptotically match the free Laplacian gap.

Different potential properties lead to distinct asymptotic behaviors.

Results depend on the nature of the underlying potential.

Abstract

We study the asymptotic behaviour of the spectral gap of Schr\"odinger operators in two and higher dimensions and in a limit where the volume of the domain tends to infinity. Depending on properties of the underlying potential, we will find different asymptotic behaviours of the gap. In some cases the gap behaves as the gap of the free Dirichlet Laplacian and in some cases it does not.