V.Grinshpun
Pure Absolutely Continuous Spectrum
for Random Operators on l^2(Z^d) at Low Disorder
(2054K, PDF)
ABSTRACT. Absence of singular continuous component, with probability one,
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 involved is the rank-one perturbation theory
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 disorder (similar proof holds for d>4).
The new result implies, via the trace-class perturbation analysis,
Anderson model with the unbounded random potential
having only pure point spectrum
(complete system of localized wave-functions) with probability one
in arbitrary dimension.
The basic idea is to establish absence of the mixed, point and continuous,
spectra in the range of the conductivity spectral component of the arbitrary
(bounded non-random) perturbation, it had been understood by author (1999)
while independent study of the exactly solvable
model, and of the disordered surface model
(explicitly considered in the paper).
Various generalizations are applicable to describe the spectral
properties of multidimensional Hamiltonians with Anderson-type potentials,
random and non-random as well
(subject to the possible forthcoming communication by the author).
The new results imply the non-zero value of conductivity
in the energy regime corresponding to the high impurity concentration
and zero temperature (at low disorder),
providing rigorous proof for the so-called Mott conjecture.