TL;DR
This paper establishes Wegner estimates for discrete alloy-type random Schrödinger operators, providing bounds on eigenvalue counts that facilitate localization proofs in quantum physics.
Contribution
It proves Wegner estimates under specific conditions on the potential and distribution, advancing the mathematical understanding of localization in disordered systems.
Findings
Wegner estimates hold for compactly supported potentials with bounded variation distributions.
The bounds are polynomial in the volume, enabling multiscale analysis.
Results support localization proofs in discrete alloy-type models.
Abstract
We study discrete alloy-type random Schr\"odinger operators on . Wegner estimates are bounds on the average number of eigenvalues in an energy interval of finite box restrictions of these types of operators. If the single site potential is compactly supported and the distribution of the coupling constant is of bounded variation a Wegner estimate holds. The bound is polynomial in the volume of the box and thus applicable as an ingredient for a localisation proof via multiscale analysis.
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