Spectral Dependence of Degree of Localization of Eigenfunctions of the 1D Schrodinger Equation with a Peacewise-Constant Random Potential

Abstract

The perturbation theory is developed for joint statistics of the advanced and retarded Green's functions of the 1D Schrodinger equation with a piecewise-constant random potential. Using this method, analytical expressions are obtained for spectral dependence of the degree of localization and for the limiting (at $t\rightarrow\infty$) probability to find the particle at the point it was located at $t = 0$ (Andeson criterion). Definition of the localization length is introduced. The computer experiments confirming correctness of the calculations are described.