Solutions, Spectrum, and Dynamica for Schr\"odinger Operators on Infinite Domains

TL;DR

This paper investigates the relationship between the growth of solutions to Schr"odinger equations on unbounded domains and the spectral and dynamical properties of the operators, providing new criteria and insights into quantum transport phenomena.

Contribution

It introduces new criteria linking growth rates of solutions to spectral measures and dynamical properties of Schr"odinger operators on infinite domains.

Findings

Established relations between solution growth and spectral measure continuity.

Derived criteria for spectral property identification.

Applied results to analyze quantum transport properties.

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.