27 research outputs found
Localization for random operators with non-monotone potentials with exponentially decaying correlations
I consider random Schr\"odinger operators with exponentially decaying single
site potential, which is allowed to change sign. For this model, I prove
Anderson localization both in the sense of exponentially decaying
eigenfunctions and dynamical localization. Furthermore, the results imply a
Wegner-type estimate strong enough to use in classical forms of multi-scale
analysis
Stability of the Periodic Toda Lattice in the Soliton Region
We apply the method of nonlinear steepest descent to compute the long-time
asymptotics of the periodic (and slightly more generally of the quasi-periodic
finite-gap) Toda lattice for decaying initial data in the soliton region. In
addition, we show how to reduce the problem in the remaining region to the
known case without solitons.Comment: 28 page
Relative Oscillation Theory for Sturm-Liouville Operators Extended
We extend relative oscillation theory to the case of Sturm--Liouville
operators with different 's. We show that the
weighted number of zeros of Wronskians of certain solutions equals the value of
Krein's spectral shift function inside essential spectral gaps.Comment: 16 page
Unique continuation for discrete nonlinear wave equations
We establish unique continuation for various discrete nonlinear wave
equations. For example, we show that if two solutions of the Toda lattice
coincide for one lattice point in some arbitrarily small time interval, then
they coincide everywhere. Moreover, we establish analogous results for the
Toda, Kac-van Moerbeke, and Ablowitz-Ladik hierarchies. Although all these
equations are integrable, the proof does not use integrability and can be
adapted to other equations as well.Comment: 10 page
Periodic and limit-periodic discrete Schr\"odinger operators
The theory of discrete periodic and limit-periodic Schr\"odinger operators is
developed. In particular, the Floquet--Bloch decomposition is discussed.
Furthermore, it is shown that an arbitrarily small potential can add a gap for
even periods. In dimension two, it is shown that for coprime periods small
potential terms don't add gaps thus proving a Bethe--Sommerfeld type statement.
Furthermore limit-periodic potentials whose spectrum is an interval are
constructed