Since the seminal work of P. Anderson in 1958, localization in
disordered systems has been the object of intense investigations.
Mathematically speaking, the phenomenon can be described as
follows: the self-adjoint operators which are used as Hamiltonians
for these systems have a ten dency to have pure point spectrum,
especially in low dimension or for large disorder. A lot of effort
has been devoted to the mathematical study of the random
self-adjoint operators relevant to the theory of localization for
disordered systems. It is fair to say that progress has been made
and that the un derstanding of the phenomenon has improved. This
does not mean that the subject is closed. Indeed, the number of
important problems actually solved is not larger than the number of
those remaining. Let us mention some of the latter: * A proof of
localization at all energies is still missing for two dimen sional
systems, though it should be within reachable range. In the case of
the two dimensional lattice, this problem has been approached by
the investigation of a finite discrete band, but the limiting pro
cedure necessary to reach the full two-dimensional lattice has
never been controlled. * The smoothness properties of the density
of states seem to escape all attempts in dimension larger than one.
This problem is particularly serious in the continuous case where
one does not even know if it is continuous.
General
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