Quantum Hamiltonians with weak random abstract perturbation. II. Localization in the expanded spectrum
arXiv:2003.03562 · doi:10.1007/s10955-020-02683-0
Abstract
We consider multi-dimensional Schrödinger operators with a weak random perturbation distributed in the cells of some periodic lattice. In every cell the perturbation is described by the translate of a fixed abstract operator depending on a random variable. The random variables, indexed by the lattice, are assumed to be independent and identically distributed according to an absolutely continuous probability density. A small global coupling constant tunes the strength of the perturbation. We treat analogous random Hamiltonians defined on multi-dimensional layers, as well. For such models we determine the location of the almost sure spectrum and its dependence on the global coupling constant. In this paper we concentrate on the case that the spectrum expands when the perturbation is switched on. Furthermore, we derive a Wegner estimate and an initial length scale estimate, which together with Combes--Thomas estimate allows to invoke the multi-scale analysis proof of localization. We specify an energy region, including the bottom of the almost sure spectrum, which exhibits spectral and dynamical localization. Due to our treatment of general, abstract perturbations our results apply at once to many interesting examples both known and new.
References in corpus (29)
- The absolute continuity of the integrated density of states for magnetic Schrödinger operators with certain unbounded random potentials
- Anderson localization for a class of models with a sign-indefinite single-site potential via fractional moment method
- Localisation for non-monotone Schroedinger operators
- Localization for the random displacement model
- Spectral extrema and Lifshitz tails for non monotonous alloy type models
- Minimizing the ground state energy of an electron in a randomly deformed lattice
- Asymptotic behaviour of the spectrum of a waveguide with distant perturbations
- Wegner estimate for discrete alloy-type models
- Wegner estimate and the density of states of some indefinite alloy type Schroedinger Operators
- Wegner estimates for sign-changing single site potentials
- Distant perturbations of the Laplacian in a multi-dimensional space
- Spectral properties of discrete alloy-type models
- Continuity of the integrated density of states on random length metric graphs
- Lifshitz tails for a class of Schrödinger operators with random breather-type potential
- Localization for the random displacement model at weak disorder
- On the Lipschitz continuity of the integrated density of states for sign-indefinite potentials
- Integrated density of states for random metrics on manifolds
- Lipschitz-continuity of the integrated density of states for Gaussian random potentials
- Continuity properties of the integrated density of states on manifolds
- Low lying eigenvalues of randomly curved quantum waveguides
- Localization via fractional moments for models on with single-site potentials of finite support
- Low lying spectrum of weak-disorder quantum waveguides
- Anderson Localization at Band Edges for Random Magnetic Fields
- The weak localization for the alloy-type Anderson model on a cubic lattice
- Quantum Hamiltonians with weak random abstract perturbation. I. Initial length scale estimate
- Expansion of the almost sure spectrum in the weak disorder regime
- Expansion of the spectrum in the weak disorder regime for random operators in continuum space
- Wegner estimate for discrete Schrödinger operators with Gaussian random potentials
- Spectral localization for quantum Hamiltonians with weak random delta interaction