Reflection probabilities of one-dimensional Schroedinger operators and scattering theory

TL;DR

This paper characterizes reflection probabilities in one-dimensional Schrödinger operators using scattering theory, showing their equivalence and providing a concise proof of a key result, inspired by non-equilibrium statistical mechanics.

Contribution

It introduces a new scattering-theoretic approach to relate dynamic and spectral reflection probabilities, simplifying proofs and extending insights from Jacobi operators.

Findings

Reflection probabilities are equal for the pair (H, H_infinity).

Provides a short, transparent proof of a known main result.

Connects scattering theory with non-equilibrium statistical mechanics.

Abstract

The dynamic reflection probability and the spectral reflection probability for a one-dimensional Schroedinger operator $H = - \Delta + V$ are characterized in terms of the scattering theory of the pair $(H, H_\infty)$ where $H_\infty$ is the operator obtained by decoupling the left and right half-lines $\mathbb{R}_{\leq 0}$ and $\mathbb{R}_{\geq 0}$. An immediate consequence is that these reflection probabilities are in fact the same, thus providing a short and transparent proof of the main result of Breuer, J., E. Ryckman, and B. Simon (2010) . This approach is inspired by recent developments in non-equilibrium statistical mechanics of the electronic black box model and follows a strategy parallel to the Jacobi case.