On the asymptotic behavior of the spectral gap for discrete Schr\"odinger operators

TL;DR

This paper investigates how the spectral gap of discrete Schrödinger operators on path graphs behaves asymptotically as the volume grows, confirming and extending previous results and resolving a conjecture.

Contribution

It generalizes existing asymptotic results for spectral gaps to a broader class of potentials using new methods and proves a conjecture in the field.

Findings

Spectral gap tends to zero at a specific rate as volume increases.

Confirmed recent asymptotic behaviors for a class of potentials.

Resolved a previously posed conjecture about spectral gaps.

Abstract

In this note we elaborate on the asymptotic behavior of the spectral gap of a class of discrete Schr\"odinger operators defined on a path graph in the limit of infinite volume. We confirm recent results and generalize them to a larger class of potentials using entirely different methods. Notably, we also resolve a conjecture previously proposed in this context. This then yields new insights into the rate at which the spectral gap tends to zero as the volume increases.