Schrödinger Operators in a Strip

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.