We prove that the integrated density of states (IDS) of random Schrödinger operators with Anderson-type potentials on L 2 (R d ), for d ≥ 1, is locally Hölder continuous at all energies with the same Hölder exponent 0 < α ≤ 1 as the conditional probability measure for the single-site random variable. As a special case, we prove that if the probability distribution is absolutely continuous with respect to Lebesgue measure with a bounded density, then the IDS is Lipschitz continuous at all energies. The single-site potential u ∈ L ∞ 0 (R d ) must be nonnegative and compactly-supported. The unperturbed Hamiltonian must be periodic and satisfy a unique continuation principle. We also prove analogous continuity results for the IDS of random Anderson-type perturbations of the Landau Hamiltonian in two-dimensions. All of these results follow from a new Wegner estimate for local random Hamiltonians with rather general probability measures.
We study the spectrum of random Schrodinger operators acting on L 2 (IR d ) of the following type H = -A + W + Σxez dt χ V χ τhe (tχ)χez' are i-i-drandom variables. Under weak assumptions on F, we prove exponential localization for H at the lower edge of its spectrum. In order to do this, we give a new proof of the Wegner estimate that works without sign assumptions on V.Resume: Dans ce travail, nous etudions le spectre d'operateurs de Schrodinger aleatoires agissant sur L 2 (IR d ) du type suivant H = -A + W + Y JXeΈ ^xV x . Les (tχ)xez d sont des variables aleatoires i.i.d. Sous de faibles hypotheses sur F, nous demontrons que le bord inferieur du spectre de H n'est compose que de spectre purement ponctuel et, que les fonctions propres associees sont exponentiellement decroissantes. Pour ce faire nous donnons une nouvelle preuve de Γestimee de Wegner valable sans hypotheses de signe sur V.
We study the integrated density of states of random Anderson-type additive and multiplicative perturbations of deterministic background operators for which the single-site potential does not have a fixed sign. Our main result states that, under a suitable assumption on the regularity of the random variables, the integrated density of states of such random operators is locally Ho¨lder continuous at energies below the bottom of the essential spectrum of the background operator for any nonzero disorder, and at energies in the unperturbed spectral gaps, provided the randomness is sufficiently small. The result is based on a proof of a Wegner estimate with the correct volume dependence. The proof relies upon the L p -theory of the spectral shift function for p51 (Comm. Math. Phys. 218 (2001), 113-130), and the vector field methods of Klopp (Comm. Math. Phys. 167 (1995), 553-569). We discuss the application of this result to Schro¨dinger operators with random magnetic fields and to band-edge localization. # 2002 Elsevier Science (USA)
We study various statistics related to the eigenvalues and eigenfunctions of random Hamiltonians in the localized regime. Consider a random Hamiltonian at an energy E in the localized phase. Assume the density of states function is not too flat near E. Restrict it to some large cube . Consider now I , a small energy interval centered at E that asymptotically contains infintely many eigenvalues when the volume of the cube grows to infinity. We prove that, with probability one in the large volume limit, the eigenvalues of the random Hamiltonian restricted to the cube inside the interval are given by independent identically distributed random variables, up to an error of size an arbitrary power of the volume of the cube. As a consequence, we derive• uniform Poisson behavior of the locally unfolded eigenvalues, • a.s. Poisson behavior of the joint distributions of the unfolded energies and unfolded localization centers in a large range of scales, • the distribution of the unfolded level spacings, locally and globally,• the distribution of the unfolded localization centers, locally and globally.
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