Lifshits asymptotics for one-dimensional stochastic operators

Abstract

I. M. Lifshits proposed [2], and then with students developed in detail [3] themethod of "optical fluctuation," connected the problem of the computation of N(I) with the solution of an auxiliary variational problem. A significant number of concrete problems both onedimensional and multidimensional were solved by this method in [3]. However, a strict mathematical basis of part of the results [2, 3], presented in a familiar series of articles of Donsker and Varadhan [4-6], does not yield information concerning the construction of "optimal potential holes" and involves cumbersome computations. This is characteristic of all methods, where the function N(%) itself is not studied, but its Laplace transform and corresponding semigroup of operators. The ideology of [2-6] is not directly tied to dimensionality. It is well known, however, that in the one-dimensional situation there are quite effective direct methods of analyzing N(1), based on the phase formalism. L. A. Pastur has actively popularized them. The object of this paper is to give a general asymptotic formula for N(~) in the class of one-dimensional Schrodinger operators with a nonnegative random potential. The answer is formulated in terms of the probabilities of large deviations of an integral mean potential, for the analysis of which there are well-developed analytic methods.