Wegner Estimate | ScienceGate

Abstract We prove a Wegner estimate for discrete Schrödinger operators with a potential given by a Gaussian random process. The only assumption is that the covariance function decays exponentially; no monotonicity assumption is required. This improves earlier results where abstract conditions on the conditional distribution, compactly supported and non-negative, or compactly supported covariance functions with positive mean are considered.

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The Wegner estimate and the integrated density of states for some random operators

Proceedings - Mathematical Sciences ◽

10.1007/bf02829639 ◽

2002 ◽

Vol 112 (1) ◽

pp. 31-53 ◽

Cited By ~ 11

Author(s):

J. M. Combes ◽

P. D. Hislop ◽

Frédéric Klopp ◽

Shu Nakamura

Keyword(s):

Density Of States ◽

Integrated Density Of States ◽

Random Operators ◽

Wegner Estimate

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Wegner Estimate for Random Divergence-Type Operators Monotone in the Randomness

Mathematical Physics Analysis and Geometry ◽

10.1007/s11040-021-09396-0 ◽

2021 ◽

Vol 24 (3) ◽

Author(s):

Alexander Dicke

Keyword(s):

Linear Dependence ◽

Unique Continuation ◽

Random Model ◽

Random Perturbations ◽

Random Parameters ◽

Wegner Estimate ◽

Non Linear ◽

Divergence Type

AbstractIn this note, a Wegner estimate for random divergence-type operators that are monotone in the randomness is proven. The proof is based on a recently shown unique continuation estimate for the gradient and the ensuing eigenvalue liftings. The random model which is studied here contains quite general random perturbations, among others, some that have a non-linear dependence on the random parameters.

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Wegner estimate and localization for alloy-type models with sign-changing exponentially decaying single-site potentials

Reviews in Mathematical Physics ◽

10.1142/s0129055x15500075 ◽

2015 ◽

Vol 27 (04) ◽

pp. 1550007 ◽

Cited By ~ 2

Author(s):

Karsten Leonhardt ◽

Norbert Peyerimhoff ◽

Martin Tautenhahn ◽

Ivan Veselić

Keyword(s):

Multiscale Analysis ◽

Step Function ◽

Single Site ◽

Energy Interval ◽

Wegner Estimate ◽

Alloy Type ◽

Fixed Sign ◽

Wegner Estimates ◽

Single Site Potential ◽

The Continuum

We study Schrödinger operators on L2(ℝd) and ℓ2(ℤd) with a random potential of alloy-type. The single-site potential is assumed to be exponentially decaying but not necessarily of fixed sign. In the continuum setting, we require a generalized step-function shape. Wegner estimates are bounds on the average number of eigenvalues in an energy interval of finite box restrictions of these types of operators. In the described situation, a Wegner estimate, which is polynomial in the volume of the box and linear in the size of the energy interval, holds. We apply the established Wegner estimate as an ingredient for a localization proof via multiscale analysis.

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