2003/08/31 by Michael Aizenman, Alexander Elgart, Serguei Naboko +4
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics #cond-mat.dis-nn #math-ph #math.MP #math.SP #msc:46N50 #msc:82B44
paper · pdf · doi:10.1007/s00222-005-0463-y
published as Inventiones Mathematicae, v. 163, p. 343 (2006). · Latex file, 63 pp; v2. introduction rewritten and other sections revised to streamline and clarify presentation; v3. a number of typos corrected
openalex publication_date 2005/10/25 · arxiv created 2005/11/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
We study localization effects of disorder on the spectral and dynamical properties of Schroedinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the analysis of fractional moments of the resolvent, which are finite due to the resonance-diffusing effects of the disorder. The main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the Lifshitz-Krein spectral shift associated with the local potential terms. The difficulty is avoided here through the use of a weak-L1 estimate concerning the boundary-value distribution of resolvents of maximally dissipative operators, combined with standard tools of relative compactness theory.