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.