Dynamical Properties of Quasi-periodic Schrödinger Equations

by Bjerklöv, Kristian

This thesis deals with the investigation of dynamical properties of quasiperiodic Schrödinger equations. It contains the following two papers: Paper I. Positive Lyapunov exponent for a class of 1-D quasiperiodic Schrödinger equations — the discrete case. For a nonconstant C1 potential function V : T ? R and for large ?, we prove that for an almost full measure set of irrational frequencies ? ? T and for a large set of energies E ? R, all lying in the spectrum of the Schrödinger operator (H?u)n = ?(un+1 + un?1) + ?V (? + n?)un, the (maximal) Lyapunov exponent associated with the equation H?u = Eu, is positive. Moreover, for these energies, the projective flow corresponding to the fundamental solution of the system ( ) ( ) ( ) un 0 1 un?1 = , ?1 ?V (? + n?) ? E un+1 is shown to be minimal. Paper II. Positive Lyapunov exponent for a class of 1-D quasiperiodic Schrödinger equations — the continuum case. We prove that in the bottom of the spectrum of the Schrödinger operator (H?u)(t) = ? d2 u(t) + ?V (t, ? + ?t)u(t), dt2 the Lyapunov exponent is positive for a large set of energies E and frequencies ? ? R \ Q, provided that ? is large and that the potential function V : T2 ? R satisfies some regularity conditions. We also prove that the projective flow corresponding to the fundamental solution of the system ( u u? )? ( ) ( 0 1 u = ?V (t, ? + ?t) ? E 0 u? ) , obtained from the Schrödinger equation H?u = Eu, is minimal in these cases. un Key Words: Schrödinger equations, Schrödinger operators, positive Lyapunov exponents, invariant measures, minimality, measure of spectrum. Mathematics Subject Classification (MSC 2000): Primary 34Cxx, 34L40, 37Cxx, 37Nxx. iii