Lectures on random Schrödinger operators

Abstract

Abstract. These notes are based on lectures given in the IV Escuela de Verano en Análisis y Fisica Matemática at the Instituto de Matemáticas, Cuernavaca, Mexico, in May 2005. It is a complete description of Anderson localization for random Schrödinger operators on L 2 (IR d ) and basic properties of the integrated density of states for these operators. A discussion of localization for randomly perturbed Landau Hamiltonians and their relevance to the integer quantum Hall effect completes the lectures.

Spectra of random operators with absolutely continuous integrated density of states

“…In [6] random Schrödinger operators of the form H ω (λ) = H 0 + λV ω on L 2 (R d ) are considered for λ ∈ R. The operator H 0 = (−i∇ − A 0 ) 2 + V 0 is nonrandom. The random Anderson type potential V ω is constructed from the nonzero single-site potential u ≥ 0 as…”

“…Therefore the IDS is absolutely continuous and we can apply Theorem 4 as mentioned above. In [6] it is shown that under some other hypothesis there is band-gap localization. This means that the spectrum close to the gaps is pure point almost surely.…”

Stochastic Regularization and Eigenvalue Concentration Bounds for Singular Ensembles of Random Operators

“…A reasonably complete review of results on EVC bounds is certainly beyond the scope of this short note. The reader can find a detailed discussion along with rich bibliography in the works [26,27].…”

On the zero-field orbital magnetic susceptibility of Bloch electrons in graphene-like solids: Some rigorous results

“…The non-degeneracy assumption for the E j 's, j = 1, 2 allows us to use the regular perturbation theory, which from the Hellman-Feynman formula (see e.g. [20,Thm 4.1]), provides the identities:…”