The local structure of the spectrum of the one-dimensional Schrödinger operator

Abstract

Let\(H_V = - \frac{{d^{\text{2}} }}{{dt^{\text{2}} }} + q(t,\omega )\) be an one-dimensional random Schrödinger operator in ℒ2(−V,V) with the classical boundary conditions. The random potentialq(t, ω) has a formq(t, ω)=F(x t ), wherex t is a Brownian motion on the compact Riemannian manifoldK andF:K→R 1 is a smooth Morse function,\(\mathop {\min }\limits_K F = 0\). Let\(N_V (\Delta ) = \sum\limits_{Ei(V) \in \Delta } 1 \), where Δ∈(0, ∞),E i (V) are the eigenvalues ofH V . The main result (Theorem 1) of this paper is the following. IfV→∞,E 0>0,k∈Z + anda>0 (a is a fixed constant) then $$P\left\{ {N_V \left( {E_0 - \frac{a}{{2V}},E_0 + \frac{a}{{2V}}} \right) = k} \right\}\xrightarrow[{V \to \infty }]{}e^{ - an(E_0 )} (an(E_0 ))^k |k!,$$ wheren(E 0) is a limit state density ofH V ,V→∞. This theorem mean that there is no repulsion between energy levels of the operatorH V ,V→∞.The second result (Theorem 2) describes the phenomen of the repulsion of the corresponding wave functions.