Estimates for Dirichlet Eigenvalues of the Schrodinger operator with the Kronig-Penney Model

TL;DR

This paper refines asymptotic formulas for Dirichlet eigenvalues of the Schrödinger operator with Kronig-Penney potentials, providing explicit estimates and numerical methods for small eigenvalues.

Contribution

It improves existing asymptotic formulas, explicitly determines terms for bounded variation potentials, and develops a fixed point iteration method for eigenvalue estimation.

Findings

Explicit asymptotic formulas for eigenvalues

Fixed point iteration for eigenvalue computation

Numerical estimates with error bounds

Abstract

In this paper, we first improve some asymptotic formulas previously obtained and provide sharp asymptotic formulas explicitly expressed by the potential. For the potentials of bounded variation, we obtain asymptotic formulas in which the first and second terms are explicitly determined and separated from the error terms. In addition, we illustrate these formulas for the Kronig-Penney potential. We then provide estimates for the small Dirichlet eigenvalues of the one-dimensional Schrodinger operator in the Kronig-Penney model. We derive several useful equations from certain iteration formulas for computing these Dirichlet eigenvalues, and prove that all the eigenvalues can be found by the fixed point iteration. Then, using the Banach fixed point theorem, we estimate the eigenvalues numerically. Moreover, we present error estimates and include a numerical example.