Modified Prufer and EFGP transforms and the spectral analysis of one-dimensional Schrodinger operators

Using control of the growth of the transfer matrices, we discuss the spectral analysis of continuum and discrete half-line Schrodinger operators with slowly decaying potentials. Among our results we show if V(x) = Sigma(n=1)(infinity)a(n)W(x - x(n)), where W has compact support and x(n)/x(n+1)-->0, then H has purely a.c. (resp. purely s.c.) spectrum on (0, infinity) if Sigma a(n)(2) < infinity (resp. Sigma a(n)(2) = infinity). For lambda n(-1/2)a(n) potentials, where a(n) are independent, identically distributed random variables with E(a(n)) = 0, E(a(n)(2)) = 1, and lambda < 2, we find singular continuous spectrum with explicitly computable fractional Hausdorff dimension.