97-195 Kiselev A., Last Y., Simon B.

Modified Pr\"ufer and EFGP Transforms and the Spectral Analysis of One-Dimensional Schr\"odinger Operators (106K, AMSTeX) Apr 9, 97

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Abstract. Using control of the growth of the transfer matrices, we discuss the spectral analysis of continuum and discrete half-line Schr\"odinger operators with slowly decaying potentials. Among our results we show if $V(x) = \sum^\infty_{n=1} a_n W(x-x_n)$ where $W$ has compact support and $x_n / x_{n+1} \to 0$, then $H$ has purely a.c.~(resp.~purely s.c.) spectrum on $(0,\infty)$ if $\sum a^2_n < \infty$ (resp.~$\sum a^2_n = \infty$). For $\lambda n^{-1/2} a_n$ potentials where $a_n$ are independent, identically distributed random variables with $E(a_n) =0$, $E(a^2_n)=1$, and $\lambda < 2$, we find singular continuous spectrum with explicitly computable fractional Hausdorff dimension.

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