Contributed by: M. Aizenman (Princeton Univ.), Dec. 15, 1998.
Establish the existence (in some energy range) of extended eigenstates,
or continuous spectrum, for linear operators
with extensive disorder.
A prototypical example is the discrete Schroedinger
operator with random potential
Delta -- the
discrete Laplacian (n.n. difference operator),
Extended states play a basic role in elementary explanations of
conduction in solid state.
The scaling theory of AALR  suggests that such states do occur
(almost surely for lambda small enough) in dimensions d > 2.
Nevertheless, despite continuing efforts, continuous spectrum (in the presence of extensive disorder) was established only for such random operators on homogeneous trees (Bethe lattices) .
A related open problem is to shed light on the situation in the borderline case (?) of d=2 dimensions.
The localized (pure-point) spectrum is better understood.
Some textbooks (with extensive reference lists) are listed under . Among the many recent localization results are extensions to other wave equations , and bounds with applications to quantum Hall systems .