Formulation of a unified method for low- and high-energy expansions in the analysis of reflection coefficients for one-dimensional Schr\"odinger equation

TL;DR

This paper introduces a unified algebraic method to derive low- and high-energy expansions of reflection coefficients for the one-dimensional Schrödinger equation, facilitating analysis of Green functions and potential behaviors.

Contribution

A generalized formalism based on the Fokker-Planck equation is developed for deriving energy expansions in a unified framework, clarifying the algebraic structure involved.

Findings

The method provides a systematic way to obtain expansions for different potential asymptotics.

The formalism enhances understanding of the algebraic structure underlying the expansions.

Validity of the expansions is examined for various asymptotic potential behaviors.

Abstract

We study low-energy expansion and high-energy expansion of reflection coefficients for one-dimensional Schr\"odinger equation, from which expansions of the Green function can be obtained. Making use of the equivalent Fokker-Planck equation, we develop a generalized formulation of a method for deriving these expansions in a unified manner. In this formalism, the underlying algebraic structure of the problem can be clearly understood, and the basic formulas necessary for the expansions can be derived in a natural way. We also examine the validity of the expansions for various asymptotic behaviors of the potential at spatial infinity.