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

الأصل · EN

Let H be a Schrödinger operator defined on an unbounded domain D in Rᵈ with Dirichlet boundary conditions (D may equal Rᵈ in particular). Let u(x,E) be a solution of the Schrödinger equation (H-E)u(x,E)=0, and let Bᵣ denote a ball of radius R centered at zero. We show relations between the rate of growth of the L² norm u(x,E)ₗ₂₍Bᵣ ∩ 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 L² 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 Schrödinger operators.