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 $t\to \infty$) 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.

18 pages, 2 figures

References in corpus (1)

• Spectrum and eigen functions of the operator $H_{U}f(x)\equiv f(U-1/x)/x^2$ and strange attractor's density for the mapping $x_{n+1}=1/(U-x_n)$