Analysis of reflection coefficients for the Fokker-Planck equation

TL;DR

This paper investigates the mathematical properties of reflection coefficients in the one-dimensional Fokker-Planck equation, introducing a new differential operator formalism to derive high- and low-energy expansions and analyze their validity.

Contribution

A novel formalism using differential operators is developed to analyze reflection coefficients in the Fokker-Planck equation across energy regimes.

Findings

Formulas for high-energy expansions derived

Formulas for low-energy expansions derived

Conditions for the validity of expansions discussed

Abstract

Mathematical structure of the reflection coefficients for the one-dimensional Fokker-Planck equation is studied. A new formalism using differential operators is introduced and applied to the analysis in high- and low-energy regions. Formulas for high-energy and low-energy expansions are derived, and expressions for the coefficients of the expansion, as well as the remainder terms, are obtained for general forms of the potential. Conditions for the validity of these expansions are discussed on the basis of the analysis of the remainder terms.