Random Sturm Liouville Operators

TL;DR

This paper investigates the spectral properties of random Sturm-Liouville operators, demonstrating that for almost all realizations, these operators do not share eigenvalues with a broad class of related random operators, using positivity conditions.

Contribution

It introduces a novel approach using positivity conditions to show eigenvalue disjointness for a broad family of random Sturm-Liouville operators.

Findings

Almost surely, $H_\omega$ does not share eigenvalues with similar operators.

The method applies to operators on subintervals of the original domain.

Eigenvalue disjointness holds for a broad class of random potentials.

Abstract

Selfadjoint Sturm-Liouville operators $H_\omega$ on $L_2(a,b)$ with random potentials are considered and it is proven, using positivity conditions, that for almost every $\omega$ the operator $H_\omega$ does not share eigenvalues with a broad family of random operators and in particular with operators generated in the same way as $H_\omega$ but in $L_2(\tilde a,\tilde b)$ where $(\tilde a,\tilde b)\subset(a,b)$.