Quantum Hamiltonians with Weak Random Abstract Perturbation. II. Localization in the Expanded Spectrum

We consider multi-dimensional Schr\"odinger operators with a weak random perturbation distributed in the cells of some periodic lattice. In every cell is described by translate fixed abstract operator depending on variable. The variables, indexed lattice, are assumed to be independent and identically according an absolutely continuous probability density. A small global coupling constant tunes strength perturbation. treat analogous Hamiltonians defined layers, as well. For such models we determine location almost sure spectrum its dependence constant. this paper concentrate case that expands when switched on. Furthermore, derive Wegner estimate initial length scale estimate, which together Combes--Thomas allows invoke multi-scale analysis proof localization. specify energy region, including bottom spectrum, exhibits spectral dynamical Due our treatment general, perturbations results apply at once many interesting examples both known new.