Localization for alloy-type models with non-monotone potentials [article]

Martin Tautenhahn
2012 arXiv   pre-print
We consider a family of self-adjoint operators [H_ω = - Δ + λ V_ω, ω∈Ω = _k ∈^d,] on the Hilbert space ℓ^2 (^d) or L^2 (^d). Here Δ denotes the Laplace operator (discrete or continuous), V_ω is a multiplication operator given by the function V_ω (x) = ∑_k ∈^dω_k u(x-k) on ^d, or V_ω (x) = ∑_k ∈^dω_k U(x-k) on ^d, and λ > 0 is a real parameter modeling the strength of the disorder present in the model. The functions u:^d → and U:^d → are called single-site potential. Moreover, there is a
more » ... ity measure on Ω modeling the distribution of the individual configurations ω∈Ω. The measure = ∏_k ∈^dμ is a product measure where μ is some probability measure on satisfying certain regularity assumptions. The operator on L^2 (^d) is called alloy-type model, and its analogue on ℓ^2 (^d) discrete alloy-type model. This thesis refines the methods of multiscale analysis and fractional moments in the case where the single-site potential is allowed to change its sign. In particular, we develop the fractional moment method and prove exponential localization for the discrete alloy-type model in the case where the support of u is finite and u has fixed sign at the boundary of its support. We also prove a Wegner estimate for the discrete alloy-type model in the case of exponentially decaying but not necessarily finitely supported single-site potentials. This Wegner estimate is applicable for a proof of localization via multiscale analysis.
arXiv:1211.3891v1 fatcat:bkj5moseq5b2jkk7mii3pbfeke