Spatial asymptotics of Green's function and applications

TL;DR

This paper investigates the behavior of Green's function for a one-dimensional Schrödinger operator with decaying potential, deriving entropy bounds on spectral measures and demonstrating the existence of absolutely continuous spectrum.

Contribution

It provides new bounds on spectral measure entropy and links these bounds to the presence of absolutely continuous spectrum in the operator.

Findings

Derived bounds on the entropy of spectral measures.

Established the presence of absolutely continuous spectrum.

Analyzed spatial asymptotics of Green's function for operator-valued potentials.

Abstract

We study the spatial asymptotics of Green's function for the 1d Schrodinger operator with operator-valued decaying potential. The bounds on the entropy of the spectral measures are obtained. They are used to establish the presence of a.c. spectrum