WKB and Spectral Analysis¶of One-Dimensional Schrödinger Operators¶with Slowly Varying Potentials
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Consider a Schrödinger operator on L2 of the line, or of a half line with appropriate boundary conditions. If the potential tends to zero and is a finite sum of terms, each of which has a derivative of some order in L1+Lp for some exponent p<2, then an essential support of the the absolutely continuous spectrum equals ℝ+. Almost every generalized eigenfunction is bounded, and satisfies certain WKB-type asymptotics at infinity. If moreover these derivatives belong to Lp with respect to a weight |x|γ with γ >0, then the Hausdorff dimension of the singular component of the spectral measure is strictly less than one.
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