9 research outputs found
The Ten Martini Problem
We prove the conjecture (known as the ``Ten Martini Problem'' after Kac and
Simon) that the spectrum of the almost Mathieu operator is a Cantor set for all
non-zero values of the coupling and all irrational frequencies.Comment: 31 pages, no figure
A Nonperturbative Eliasson's Reducibility Theorem
This paper is concerned with discrete, one-dimensional Schr\"odinger
operators with real analytic potentials and one Diophantine frequency. Using
localization and duality we show that almost every point in the spectrum admits
a quasi-periodic Bloch wave if the potential is smaller than a certain constant
which does not depend on the precise Diophantine conditions. The associated
first-order system, a quasi-periodic skew-product, is shown to be reducible for
almost all values of the energy. This is a partial nonperturbative
generalization of a reducibility theorem by Eliasson. We also extend
nonperturbatively the genericity of Cantor spectrum for these Schr\"odinger
operators. Finally we prove that in our setting, Cantor spectrum implies the
existence of a -set of energies whose Schr\"odinger cocycle is not
reducible to constant coefficients
Essential spectra and exponential estimates of eigenfunctions of lattice operators of quantum mechanics
This paper is devoted to estimates of the exponential decay of eigenfunctions
of difference operators on the lattice Z^n which are discrete analogs of the
Schr\"{o}dinger, Dirac and square-root Klein-Gordon operators. Our
investigation of the essential spectra and the exponential decay of
eigenfunctions of the discrete spectra is based on the calculus of so-called
pseudodifference operators (i.e., pseudodifferential operators on the group
Z^n) with analytic symbols and on the limit operators method. We obtain a
description of the location of the essential spectra and estimates of the
eigenfunctions of the discrete spectra of the main lattice operators of quantum
mechanics, namely: matrix Schr\"{o}dinger operators on Z^n, Dirac operators on
Z^3, and square root Klein-Gordon operators on Z^n