Positive Lyapunov exponent and minimality for a class of one-dimensional quasi-periodic Schrodinger equations
2005 (English)In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 25, 1015-1045 p.Article in journal (Refereed) Published
We study the discrete quasi-periodic Schrodinger equation -(u(n+1) + u(n-1)) + lambda V(theta + n omega)u(n) = Eu-n with a non-constant C-1 potential function V : T -> R. We prove that for sufficiently large k there is a set Omega subset of T of frequencies omega, whose measure tends to 1 as lambda -> infinity, with the following property. For each w e Q there is a 'large' (in measure) set of energies E, all lying in the spectrum of the associated Schrodinger operator (and hence giving a lower estimate on the measure of the spectrum), such that the Lyapunov exponent is positive and, moreover, the projective dynamical system induced by the Schrodinger cocycle is minimal but not ergodic.
Place, publisher, year, edition, pages
2005. Vol. 25, 1015-1045 p.
IdentifiersURN: urn:nbn:se:kth:diva-14974DOI: 10.1017/s0143385704000999ISI: 000231201400003ScopusID: 2-s2.0-23444433597OAI: oai:DiVA.org:kth-14974DiVA: diva2:333015
QC 201005252010-08-052010-08-05Bibliographically approved