Random Sturm–Liouville operators

Self-adjoint Sturm–Liouville operators H ω on L 2 ( a , b ) with random potentials are considered and it is proven, using positivity conditions, that for almost every ω the operator H ω does not share eigenvalues with a broad family of random operators and in particular with operators generated in the same way as H ω but in L 2 ( a ̃ , b ̃ ) where ( a ̃ , b ̃ ) ⊂ ( a , b ) .