- 99-206 A. Kiselev, Y. Last
- Solutions, spectrum, and dynamics for Schr\"odinger operators on
infinite domains (revised)
(345K, Postscript)
Jun 2, 99
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Abstract. 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.
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