Random Sturm-Liouville operators

Self-adjoint Sturm-Liouville operators Hω on L2(a,b) with random potentials are considered and it is proven, using positivity conditions, that for almost every ω the operator Hω does not share eigenvalues with a broad family of random operators and in particular with operators generated in the same way as Hω but in L2(ã,b̃) where (ã,b̃)⊂(a,b).