Semiclassical analysis of low and zero energy scattering for one-dimensional Schrödinger operators with inverse square potentials

Ovidiu Costin, Wilhelm Schlag, Wolfgang Staubach, Saleh Tanveer

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4 Citations (Scopus)

Abstract

This paper studies the scattering matrix S (E ; ?) of the problem- ?2 ?? (x) + V (x) ? (x) = E ? (x) for positive potentials V ? C8 (R) with inverse square behavior as x ? ± 8. It is shown that each entry takes the form Si j (E ; ?) = Si j(0) (E ; ?) (1 + ? si j (E ; ?)) where Si j(0) (E ; ?) is the WKB approximation relative to the modified potentialV (x) + frac(?2, 4) <x >-2 and the correction terms si j satisfy | ?Ek si j (E ; ?) | = Ck E- k for all k = 0 and uniformly in (E, ?) ? (0, E0) × (0, ?0) where E0, ?0 are small constants. This asymptotic behavior is not universal: if - ?2 ?x2 + V has a zero energy resonance, then S (E ; ?) exhibits different asymptotic behavior as E ? 0. The resonant case is excluded here due to V > 0. © 2008 Elsevier Inc. All rights reserved.

Original languageEnglish
Pages (from-to)2321-2362
Number of pages42
JournalJournal of Functional Analysis
Volume255
Issue number9
DOIs
Publication statusPublished - 1 Nov 2008

Keywords

  • Inverse square potential
  • Modified WKB
  • Scattering matrix
  • Schrödinger operators
  • Zero energy scattering

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