Spectral gaps for self-adjoint second order operators

TL;DR

This paper establishes explicit lower bounds on the spectral gap between the two lowest eigenvalues of a perturbed second order self-adjoint operator with mixed boundary conditions, depending on domain geometry and operator coefficients.

Contribution

It provides a novel explicit lower bound on the spectral gap for perturbed self-adjoint operators with mixed boundary conditions, considering geometric and coefficient dependencies.

Findings

Derived explicit lower bounds on spectral gaps.

Analyzed dependence of bounds on domain and operator parameters.

Applied bounds to various asymptotic regimes and examples.

Abstract

We consider a second order self-adjoint operator in a domain which can be bounded or unbounded. The boundary is partitioned into two parts with Dirichlet boundary condition on one of them, and Neumann condition on the other. We assume that the potential part of this operator is non-negative. We add a localized perturbation assuming that it produces two negative isolated eigenvalues being the two lowest spectral values of the resulting perturbed operator. The main result is a lower bound on the gap between these two eigenvalues. It is given explicitly in terms of the geometric properties of the domain and the coefficients of the perturbed operator. We apply this estimate to several asymptotic regimes studying its dependence on various parameters. We discuss specific examples of operators to which the bounds can be applied.