3D Schr\"odinger equation: scattering operator, scattering amplitude and ergodic property

TL;DR

This paper explores the relationship between stationary and dynamical scattering in the 3D Schrödinger equation, establishing a quantum analog of classical ergodic formulas and connecting scattering amplitude with the scattering operator.

Contribution

It introduces a simple interconnection between the scattering amplitude and the scattering operator, linking stationary and dynamical scattering problems in quantum mechanics.

Findings

Established a link between scattering amplitude and scattering operator.

Provided a quantum analog of classical ergodic formulas.

Analyzed stationary and dynamical scattering in 3D Schrödinger equation.

Abstract

Stationary scattering problem (when the distance $r$ tends to infinity) and dynamical scattering problem (when the time $t$ tends to infinity) are considered for the 3D Schr\"odinger equation. A simple interconnection between the scattering amplitude (stationary case) and scattering operator (dynamical case) is given in the paper. This result is a quantum mechanical analog of the ergodic formulas in the classical mechanics.