Low-energy expansion formula for one-dimensional Fokker-Planck and Schr\"odinger equations with asymptotically periodic potentials

TL;DR

This paper extends the low-energy expansion method to analyze the Green function behavior for one-dimensional Fokker-Planck and Schrödinger equations with potentials that become periodic at infinity, providing new insights into their low-energy properties.

Contribution

The paper introduces an extension of the low-energy expansion technique to asymptotically periodic potentials in one-dimensional equations, broadening its applicability.

Findings

Derived low-energy Green function expansion for asymptotically periodic potentials

Identified asymptotic behavior of solutions at low energy

Provided a framework for analyzing similar differential equations

Abstract

We consider one-dimensional Fokker-Planck and Schr\"odinger equations with a potential which approaches a periodic function at spatial infinity. We extend the low-energy expansion method, which was introduced in previous papers, to be applicable to such asymptotically periodic cases. Using this method, we study the low-energy behavior of the Green function.