On the basic states of one-dimensional disordered structures

The purpose of this paper is to study a limit probability distribution of the set of the first κ eigenvalues λ₁(?)<λ₂(?)<...<λ_κ(?) (with a fixed κ and ?→∞) of the boundary problem on the interval 0, ?] wherea(t, ω),q(t, ω) are the random stationary processes. Particularly the question of the repulsion between the first eigenvalues (small energetic levels) is studied. It has been proved that in the “divergent” case (q(t, ω)=0,a(t, ω)≠0) levels repulsion exists. As for the “potential” case (a(t, ω)≡1,q(t, ω)≠0) there is not any repulsion at all. This is one of the main differences between these two cases.