Inventiones mathematicae

, 163:343

Moment analysis for localization in random Schrödinger operators


DOI: 10.1007/s00222-005-0463-y

Cite this article as:
Aizenman, M., Elgart, A., Naboko, S. et al. Invent. math. (2006) 163: 343. doi:10.1007/s00222-005-0463-y


We study localization effects of disorder on the spectral and dynamical properties of Schrödinger 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.

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© Springer-Verlag 2005

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Department of MathematicsStanford UniversityUSA
  3. 3.Department of MathematicsSt. Petersburg State UniversitySt. PetersburgRussia
  4. 4.Theoretische PhysikETH ZürichZürichSwitzerland
  5. 5.Department of MathematicsUniversity of AlabamaBirminghamUSA