Reflectionless Sturm-Liouville Equations

TL;DR

This paper investigates the spectral properties of perturbed periodic Sturm-Liouville equations, showing that zero reflection coefficient implies unitary equivalence of the operators, with detailed estimates for related functions.

Contribution

It establishes a rigorous link between zero reflection coefficient and unitary equivalence in Sturm-Liouville operators with compactly supported perturbations.

Findings

Zero reflection coefficient implies unitary equivalence of operators.

Provides bounds and asymptotics for m-functions.

Analyzes scattering coefficients in perturbed periodic equations.

Abstract

We consider compactly supported perturbations of periodic Sturm-Liouville equations. In this context, one can use the Floquet solutions of the periodic background to define scattering coefficients. We prove that if the reflection coefficient is identically zero, then the operators corresponding to the periodic and perturbed equations, respectively, are unitarily equivalent. In some appendices, we also provide the proofs of several basic estimates, e.g. bounds and asymptotics for the relevant m-functions.