- 06-171 V.Grinshpun
- Anderson Model and Absence of Pure Singular Spectrum
Jun 1, 06
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Abstract. Absence of singular continuous component, with probability 1, in the spectra of random perturbations of multidimensional finite-difference Hamiltonians, is for the first time rigorously established under certain conditions ensuring either absence of point component, or absence of absolutely continuous component in the corresponding regions of spectra. The main technical tool applied is the theory of rank-one perturbations of singular spectra.
The respective new result (the non-mixing property) is applied to establish existence and bounds of the (non-empty) pure absolutely continuous component in the spectrum of the Anderson model with bounded random potential in dimension d=2 at low dosorder. The same proof holds for d>4.
The new (1999) result implies, via the trace-class perturbation analysis, the Anderson model with the unbounded random potential to have only pure point spectrum (complete system of localized wave-functions)with probability one in arbitrary dimension.
The original approximation scheme, based on the resolvent reduction formula, and analogue of the Lippman-Schwinger equations for the generalized eigenfunctions, is applicable in order to establish existence of the non-empty pure absolutely continuous spectral component in the range of the conductivity spectrum of the arbitrary bounded perturbation in the exactly solvable model, in the surface model, and of some other multidimensional Hamiltonians, random and non-random as well.
The new original results imply non-zero value of conductivity at low disorder and zero temperature, providing rigorous proof for the Mott conjecture (on existence of localization transition, from the metal-type diffusion, corresponding to the non-empty pure absolutely continuous spectral component at low disorder, to the quantum jump diffusion corresponding to the pure point spectrum at high disorder).