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

TL;DR

This paper derives a low-energy expansion formula for the Green function of one-dimensional Fokker-Planck and Schrödinger equations with periodic potentials, aiding in understanding their spectral properties at low energies.

Contribution

It introduces a novel power series expansion formula for reflection coefficients in terms of wave number, applicable to low-energy analysis of these equations.

Findings

Derived a low-energy expansion formula for the Green function

Provided a systematic method for reflection coefficient expansion

Enhanced understanding of spectral behavior at low energies

Abstract

We study the low-energy behavior of the Green function for one-dimensional Fokker-Planck and Schr\"odinger equations with periodic potentials. We derive a formula for the power series expansion of reflection coefficients in terms of the wave number, and apply it to the low-energy expansion of the Green function.