On Explicit Estimations for the Bloch Eigenvalues of the One-dimensional Schr\"odinger Operator and the Kronig-Penney Model

TL;DR

This paper derives explicit asymptotic formulas and estimates for eigenvalues of the one-dimensional Schrödinger operator with periodic potential, with applications to the Kronig-Penney model, including numerical validation.

Contribution

It provides explicit asymptotic expansions for eigenvalues and estimates for spectral gaps in the Kronig-Penney model, enhancing understanding of spectral properties.

Findings

Asymptotic formulas for large eigenvalues are explicitly derived.

Estimates for small eigenvalues and spectral gaps are provided.

Numerical examples validate the theoretical results.

Abstract

In this paper, we consider the small and large eigenvalues of the one-dimensional Schr\"odinger operator L(q) with a periodic, real and locally integrable potential q. First we explicitly write out the first and second terms of the asymptotic formulas for the large periodic and antiperiodic eigenvalues and illustrate these formulas for the Kronig-Penney model. Then we give estimates for the small periodic and antiperiodic eigenvalues and for the length of the first gaps in the case of the Kronig-Penney model. Moreover, we give error estimations and present a numerical example.