Internal Lifshits Tails for Random Perturbations of Periodic Schrödinger Operators
نویسنده
چکیده
0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 1. Some preliminary considerations on periodic Schrödinger operators . . . . . . . . 340 1.1. The Floquet decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 1.2. Wannier basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 1.3. The density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 2. Lifshits tails behavior for random Schrödinger operators . . . . . . . . . . . . . . . . . . 343 2.1. The main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 2.2. Some applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 3. The proof of Proposition 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 4. A reduction procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 4.1. The reference operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 4.2. The reduction to a discrete problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 5. The proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 5.1. The periodic approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 5.2. Cutoff in energy for the periodic approximations . . . . . . . . . . . . . . . . . . . . . 362 6. The lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 7. The upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 7.1. Case (2) of assumption (H.2bis) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 7.2. Case (1) of assumption (H.2bis) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
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تاریخ انتشار 1999