Communications in Mathematical Physics

, Volume 218, Issue 2, pp 245–262 | Cite as

WKB and Spectral Analysis¶of One-Dimensional Schrödinger Operators¶with Slowly Varying Potentials

  • Michael Christ
  • Alexander Kiselev

Abstract:

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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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Michael Christ
    • 1
  • Alexander Kiselev
    • 2
  1. 1.Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA.¶E-mail: mchrist@math.berkeley.eduUS
  2. 2.Department of Mathematics, University of Chicago, Chicago, IL 60637, USA.¶E-mail: kiselev@math.uchicago.eduUS

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