Random Sturm-Liouville Operators with Generalized Point Interactions

TL;DR

This paper investigates the spectral properties of selfadjoint Sturm-Liouville operators with generalized point interactions, focusing on eigenvalue stability and extending Pastur's theorem to non-ergodic random interaction models.

Contribution

It introduces a framework for analyzing eigenvalues under generalized point interactions and extends Pastur's theorem to non-ergodic random cases.

Findings

Eigenvalues are stable under parameter variations of the interaction matrix.

Except for degenerate cases, energies are almost surely not eigenvalues in the random model.

Extension of Pastur's theorem to non-ergodic, independent, non-identically distributed random interactions.

Abstract

In this work we study the point spectra of selfadjoint Sturm-Liouville operators with generalized point interactions, where the two one-sided limits of the solution data are related via a general $\mathrm{SL}(2,\mathbb{R})$ matrix. We are particularly interested in the stability of eigenvalues with respect to the variation of the parameters of the interaction matrix. As a particular application to the case of random generalized point interactions we establish a version of Pastur's theorem, stating that except for degenerate cases, any given energy is an eigenvalue only with probability zero. For this result, independence is important but identical distribution is not required, and hence our result extends Pastur's theorem from the ergodic setting to the non-ergodic setting.