# Dynamical Properties of Quasi-periodic Schrödinger Equations

Abstract (Summary)

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
Bibliographical Information:

Advisor:

School:Kungliga Tekniska högskolan

School Location:Sweden

Source Type:Doctoral Dissertation

Keywords:MATHEMATICS; Schrödinger equations; Schrödinger operators; positive Lyapunov exponents; invariant measures; minimality; measure of spectrum

ISBN:91-7283-584-2

Date of Publication:01/01/2003