Random Sturm-Liouville Operators with Point Interactions

TL;DR

This paper investigates the eigenvalue invariance of selfadjoint Sturm-Liouville operators with point interactions, showing that in a probabilistic setting, points are either eigenvalues for all operators or almost none, without requiring ergodicity.

Contribution

It generalizes the eigenvalue invariance result to non-measurable, non-ergodic families of Sturm-Liouville operators with point interactions using classical oscillation theory.

Findings

A point is either an eigenvalue for all operators or for a measure-zero set.

Classical oscillation theory can determine when a point is an eigenvalue for all operators.

The results extend known ergodic operator facts to more general settings.

Abstract

We study invariance for eigenvalues of families of selfadjoint Sturm-Liouville operators with local point interactions. In a probabilistic setting, we show that a point is either an eigenvalue for all members of the family or only for a set of measure zero. Using classical oscillation theory it is possible to decide whether the second situation happens. The operators do not need to be measurable or ergodic. This generalizes the well-known fact that for ergodic operators a point is eigenvalue with probability zero.