  Let H be a Schr\"odinger operator defined on an unbounded domain D in R^d
with Dirichlet boundary conditions (D may equal R^d in particular). Let u(x,E)
be a solution of the Schr\"odinger equation (H-E)u(x,E)=0, and let B_R denote a
ball of radius R centered at zero. We show relations between the rate of growth
of the L^2 norm \|u(x,E)\|_{L^2(B_R \cap D)} of such solutions as R goes to
infinity, and continuity properties of spectral measures of the operator H.
These results naturally lead to new criteria for identification of various
spectral properties. We also prove new fundamental relations berween the rate
of growth of L^2 norms of generalized eigenfunctions, dimensional properties of
the spectral measures, and dynamical properties of the corresponding quantum
systems. We apply these results to study transport properties of some
particular Schr\"odinger operators.
