Finite-volume Criteria for Anderson Localization

For random Schrödinger operators, and a more general class of operators with random potentials of ‘regular’ probability distributions, we present a family of constructive criteria for the localization regime. A technically convenient characterization of localization is rapid decay of the Green function’s fractional moments. In addition to explicit bounds, the constructive criteria indicate that the exponential decay of the expectation values of such functions may indeed characterize the entire regime of localization. This has qualitative consequences – since the fractional moment condition is known to have other significant implications, such as dynamical, as well as spectral, localization, and the exponential decay of the expectation values of the spectral projection kernels. In the converse direction, the criteria also rule out fast power-law decay of the Green function at mobility edges. AMS subject Classification: 82B44 (Primary), 47B60, 60H25.