Random Sturm Liouville Operators

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 $(\tilde a,\tilde b)\subset(a,b)$.