## Localization for random perturbations of periodic Schrödinger operators (1996)

Venue: | RANDOM OPER. STOCHASTIC EQUATIONS |

Citations: | 59 - 20 self |

### BibTeX

@ARTICLE{Kirsch96localizationfor,

author = {Werner Kirsch and Peter Stollmann and Günter Stolz},

title = {Localization for random perturbations of periodic Schrödinger operators},

journal = {RANDOM OPER. STOCHASTIC EQUATIONS},

year = {1996},

volume = {6},

pages = {241--268}

}

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### OpenURL

### Abstract

We prove localization for Anderson-type random perturbations of periodic Schrödinger operators on R d near the band edges. General, possibly unbounded, single site potentials of fixed sign and compact support are allowed in the random perturbation. The proof is based on the following methods: (i) A study of the band shift of periodic Schrodinger operators under linearly coupled periodic perturbations. (ii) A proof of the Wegner estimate using properties of the spatial distribution of eigenfunctions of finite box hamiltonians. (iii) An improved multiscale method together with a result of de Branges on the existence of limiting values for resolvents in the upper half plane, allowing for rather weak disorder assumptions on the random potential. (iv) Results from the theory of generalized eigenfunctions and spectral averaging. The paper aims at high accessibility in providing details for all the main steps in the proof.