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Schrödinger Operators in a Strip

  • Philippe Bougerol
  • Jean Lacroix
Part of the Progress in Probability and Statistics book series (PRPR, volume 8)

Abstract

Most of the results obtained for the one dimensional Schrödinger operator in the preceding chapters can be adapted to the case of the strip. Deterministic properties are easily generalized, replacing at time, the Poincare half plane by the generalized Siegel half plane. For random operators, the theory of Lyapunov exponents is more involved and we refer heavily to part A chapter IV for related results. In contrast with the one dimensional case, positivity of some exponents is not easily checked and needs more assumptions on the “disorder”. As in chapter III we give two proofs of the pure point spectrum property. The first proof is a straightforward adaptation of Kotani’s criterion. In order to treat the Helmotz case and to obtain some information about “thermodynamic limits” we need some Laplace analysis on symplectic groups. Such results, easily obtained on SL(2,ℝ), require here much more work since we have to deal with the Poisson kernels associated to the boundaries of symplectic groups.

Keywords

Lebesgue Measure Lyapunov Exponent Haar Measure Borel Subset Symplectic Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston, Inc. 1985

Authors and Affiliations

  • Philippe Bougerol
    • 1
  • Jean Lacroix
    • 2
  1. 1.UER de MathématiquesUniversité Paris 7ParisFrance
  2. 2.Département de MathématiquesUniversité de Paris XIIIVilletaneuseFrance

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