Algebraic spectral gaps

TL;DR

This paper revisits and extends a method for systematically identifying spectral gaps in the one-dimensional Schrödinger equation, explores its generalizations, and discusses potential applications in higher dimensions.

Contribution

It offers an alternative formulation of the existing spectral gap method, proposes generalizations, and analyzes its applicability to higher-dimensional Schrödinger problems.

Findings

Alternative formulation of the spectral gap method

Proposed generalizations of the method

Discussion on higher-dimensional applications

Abstract

For the one-dimensional Schr\"odinger equation, some real intervals with no eigenvalues (the spectral gaps) may be obtained rather systematically with a method proposed by H. Giacomini and A. Mouchet in 2007. The present article provides some alternative formulation of this method, suggests some possible generalizations and extensively discusses the higher-dimensional case.