Solutions, Spectrum, and Dynamica for Schrödinger Operators on Infinite Domains

Details Solutions, Spectrum, and Dynamica for Schrödinger Operators on Infinite Domains Solutions, Spectrum, and Dynamica for Schrödinger Operators on Infinite Domains

1999-06-02

arxiv.org/abs/math/9906021v1

Mathematics

Spectral Theory

28 pages, 1 figure

Scientific paper

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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