Klopp: Localization for Schrödinger operators with random vector potentials (original) (raw)

Localization for Schrodinger operators with random vector potentials

arXiv: Mathematical Physics, 2007

We prove Anderson localization at the internal band-edges for periodic magnetic Schr{\"o}dinger operators perturbed by random vector potentials of Anderson-type. This is achieved by combining new results on the Lifshitz tails behavior of the integrated density of states for random magnetic Schr{\"o}dinger operators, thereby providing the initial length-scale estimate, and a Wegner estimate, for such models.

Anderson localization for time periodic random Schrödinger operators

Communications in Partial Differential …, 2003

We prove that at large disorder, Anderson localization in Z d is stable under localized time-periodic perturbations by proving that the associated quasi-energy operator has pure point spectrum. The formulation of this problem is motivated by questions of Anderson localization for non-linear Schrödinger equations.

Anderson localization and Lifshits tails for random surface potentials

Journal of Functional Analysis, 2006

We consider Schrödinger operators on L 2 (R d ) with a random potential concentrated near the surface R d 1 × {0} ⊂ R d . We prove that the integrated density of states of such operators exhibits Lifshits tails near the bottom of the spectrum. From this and the multiscale analysis by Boutet de Monvel and Stollmann [Arch. Math. 80 we infer Anderson localization (pure point spectrum and dynamical localization) for low energies. Our proof of Lifshits tail relies on spectral properties of Schrödinger operators with partially periodic potentials. In particular, we show that the lowest energy band of such operators is parabolic.

Moment analysis for localization in random Schrödinger operators

Inventiones mathematicae, 2006

We study localization effects of disorder on the spectral and dynamical properties of Schrödinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the analysis of fractional moments of the resolvent, which are finite due to the resonance-diffusing effects of the disorder. The main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the Lifshitz-Krein spectral shift associated with the local potential terms. The difficulty is avoided here through the use of a weak-L 1 estimate concerning the boundary-value distribution of resolvents of maximally dissipative operators, combined with standard tools of relative compactness theory.

Persistence of Anderson localization in Schrödinger operators with decaying random potentials

Arkiv for Matematik, 2007

We show persistence of both Anderson and dynamical localization in Schrödinger operators with non-positive (attractive) random decaying potential. We consider an Anderson-type Schrödinger operator with a non-positive ergodic random potential, and multiply the random potential by a decaying envelope function. If the envelope function decays slower than |x|-2 at infinity, we prove that the operator has infinitely many eigenvalues below zero. For envelopes decaying as |x|-α at infinity, we determine the number of bound states below a given energy E

Internal Lifshits tails for random magnetic Schr�dinger operators

J Funct Anal, 2007

This paper is devoted to the study of Lifshits tails for weak random magnetic perturbations of periodic Schrödinger operators acting on L 2 (R d) of the form H λ,w = (−i∇ − λ γ ∈Z d w γ A(• − γ)) 2 + V , where V is a Z d-periodic potential, λ is positive coupling constants, (w γ) γ ∈Z d are i.i.d and bounded random variables and A ∈ C 1 0 (R d , R d) is the single site vector magnetic potential. We prove that, for λ small, at an open band edge, a true Lifshits tail for the random magnetic Schrödinger operator occurs if a certain set of conditions on H 0 = − + V and on A holds.

Localization for random Schr�dinger operators with correlated potentials

Commun Math Phys, 1991

We prove localization at high disorder or low energy for lattice Schrόdinger operators with random potentials whose values at different lattice sites are correlated over large distances. The class of admissible random potentials for our multiscale analysis includes potentials with a stationary Gaussian distribution whose covariance function C(x,y) decays as \x-y\~θ, where θ>0 can be arbitrarily small, and potentials whose probability distribution is a completely analytical Gibbs measure. The result for Gaussian potentials depends on a multivariable form of Nelson's best possible hypercontractive estimate.