Analytic bounds on transmission probabilities

TL;DR

This paper introduces new analytic bounds on transmission and reflection probabilities in one-dimensional quantum scattering, enabling insights into complex potentials without requiring exact solutions.

Contribution

It presents a novel method to derive bounds on scattering probabilities by relating complex potentials to simpler, known potentials, avoiding full solutions.

Findings

Derived bounds improve understanding of transmission in complex potentials

Method applicable to a wide class of one-dimensional scattering problems

Provides a practical approach to estimate scattering properties without exact solutions

Abstract

We develop some new analytic bounds on transmission probabilities (and the related reflection probabilities and Bogoliubov coefficients) for generic one-dimensional scattering problems. To do so we rewrite the Schrodinger equation for some complicated potential whose properties we are trying to investigate in terms of some simpler potential whose properties are assumed known, plus a (possibly large) "shift" in the potential. Doing so permits us to extract considerable useful information without having to exactly solve the full scattering problem.