Lower bounds on the lowest spectral gap of singular potential Hamiltonians

TL;DR

This paper derives lower bounds on the spectral gap of Schrödinger operators with singular potentials supported on sub-manifolds, linking the gap size to geometric and spectral parameters.

Contribution

It provides new estimates on the spectral gap for singular potential Hamiltonians, especially for curves in two-dimensional space, based on geometric properties.

Findings

Spectral gap bounds depend on curve length, diameter, and curvature.

The estimates relate the gap size to geometric and spectral parameters.

Results apply to operators with at least two eigenvalues below the essential spectrum.

Abstract

We analyze Schr\"odinger operators whose potential is given by a singular interaction supported on a sub-manifold of the ambient space. Under the assumption that the operator has at least two eigenvalues below its essential spectrum we derive estimates on the lowest spectral gap. In the case where the sub-manifold is a finite curve in two dimensional Euclidean space the size of the gap depends only on the following parameters: the length, diameter and maximal curvature of the curve, a certain parameter measuring the injectivity of the curve embedding, and a compact sub-interval of the open, negative energy half-axis which contains the two lowest eigenvalues.