On the asymptotic behavior of the spectral gap for discrete Schrödinger operators

Abstract

In this note we elaborate on the asymptotic behavior of the spectral gap of a class of discrete Schrödinger 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.

The previous submission was extended to include the convergence of the spectral gap in the special case where the potential is supported only at the origin (Section 5 and an appendix added)