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

TL;DR

This paper introduces a new formalism for analyzing the high-energy asymptotic behavior of Green functions in one-dimensional Fokker-Planck and Schr"odinger equations, deriving formulas and studying their validity.

Contribution

It presents a novel formalism for high-energy asymptotic expansion of Green functions and analyzes the conditions for the expansion's validity.

Findings

Derived formulas for asymptotic expansion in inverse wave number

Studied the validity conditions via remainder term analysis

Discussed short-time expansion of the Green function

Abstract

A new formalism is presented for high-energy analysis of the Green function for Fokker-Planck and Schr\"odinger equations in one dimension. Formulas for the asymptotic expansion in powers of the inverse wave number are derived, and conditions for the validity of the expansion are studied through the analysis of the remainder term. The short-time expansion of the Green function is also discussed.