A Probabilistic Approach to One-Dimensional Schrödinger Operators with Sparse Potentials

We consider the one-dimensional Schrödinger equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} with sparse potential V (i.e.\ mainly V= 0). It is shown that the asymptotics of the solutions corresponding to positive energies E can be approximately described by an infinite sum of independent random variables. Using results from probability theory, we can then determine the spectral properties of the operators under consideration. We prove absolute continuity for a general class of potentials, and we also have examples with singular continuous spectrum.