On~the spectrum of the one-dimensional Schr\"odinger equation with a~random potential

M. M. Benderskii ; L. A. Pastur

Sbornik. Mathematics, Tome 11 (1970) no. 2, pp. 245-256

Résumé

Let $\mathfrak N(\lambda,a,b)$ be the number of eigenvalues not exceeding $\lambda$ for the selfadjoint boundary problem \begin{gather*} -y''+q(x)y=\lambda y,\\ y(a)\cos\alpha-y'(a)\sin\alpha=0,\quad y(b)\cos\beta-y'(b)\sin\beta=0 \end{gather*} with random potential $q(x)$, and let $$ N(\lambda)=\lim_{L\to\infty}\frac{\mathfrak N(\lambda,0,\,L)}L. $$ Our problem is to clarify the conditions under which this function will exist and to indicate methods for calculating it. In the present article we establish the existence of a nonrandom limit $N(\lambda)$ for a wide class of stationary ergodic potentials. This limit is calculated under the assumption that the potential $q(x)$ is Markovian, and the argument is based on the well-known theorems of Sturm. At the end of the article we consider an example in which $q(x)$ is a Markov process with two states. In this case the calculations can all be carried out completely in a practical way, with the result that we obtain a formula expressing $N(\lambda)$ by means of integrals of elementary functions. Bibliography: 9 titles.