ON THE SPECTRUM OF THE ONE-DIMENSIONAL SCHRÖDINGER EQUATION WITH A RANDOM POTENTIAL

Abstract

Let be the number of eigenvalues not exceeding for the selfadjoint boundary problem with random potential , and let 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 for a wide class of stationary ergodic potentials. This limit is calculated under the assumption that the potential 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 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 by means of integrals of elementary functions.Bibliography: 9 items.