Low-energy asymptotic expansion of the Green function for one-dimensional Fokker-Planck and Schr\"odinger equations

TL;DR

This paper introduces a new systematic method to analyze the low-energy asymptotic behavior of Green functions for one-dimensional Schrödinger and Fokker-Planck equations, enabling arbitrary order calculations and remainder analysis.

Contribution

A novel method for systematically calculating low-energy asymptotic expansions of Green functions in one-dimensional quantum and stochastic systems.

Findings

Method allows arbitrary order expansion calculation.

Remainder behavior analyzed via transmission and reflection coefficients.

Applicable to both Schrödinger and Fokker-Planck equations.

Abstract

We consider Schr\"odinger equations and Fokker-Planck equations in one dimension, and study the low-energy asymptotic behavior of the Green function using a new method. In this method, the coefficient of the expansion in powers of the wave number can be systematically calculated to arbitrary order, and the behavior of the remainder term can be analyzed on the basis of an expression in terms of transmission and reflection coefficients.