Modified Prüfer and EFGP Transforms and the Spectral Analysis of One-Dimensional Schrödinger Operators

Abstract

Using control of the growth of the transfer matrices, we discuss the spectral analysis of continuum and discrete half-line Schrödinger operators with slowly decaying potentials. Among our results we show if, where W has compact support and, then H has purely a.c. (resp. purely s.c.) spectrum on (O,∞) if). For λn^({-1/2}) ɑ_n potentials, where a n are independent, identically distributed random variables with E(ɑ_n ) = O, E(ɑ^2_n)=1, and λ < 2, we find singular continuous spectrum with explicitly computable fractional Hausdorff dimension.