A Riemann--Hilbert Problem Approach to Periodic Infinite Gap Hill's Operators and the Korteweg--de Vries Equation

TL;DR

This paper develops a Riemann--Hilbert problem framework for analyzing infinite gap Hill's operators and their evolution under the KdV equation, providing new spectral characterization and uniqueness results.

Contribution

It introduces a novel Riemann--Hilbert problem formulation for infinite gap Hill's operators and establishes uniqueness and spectral characterization results within this framework.

Findings

Established a uniqueness theorem for the Riemann--Hilbert problem

Derived a spectral characterization via scalar Riemann--Hilbert problems

Linked spectral properties to temporal periodicity under KdV evolution

Abstract

We formulate the inverse spectral theory of infinite gap Hill's operators with bounded periodic potential as a Riemann--Hilbert problem on a typically infinite collection of spectral bands and gaps. We establish a uniqueness theorem for this Riemann--Hilbert problem, which provides a new route to establishing unique determination of periodic potentials from spectral data. As the potential evolves according to the KdV equation, we use integrability to derive an associated Riemann--Hilbert problem with explicit time dependence. Basic principles from the theory of Riemann--Hilbert problems yield a new characterization of spectra for periodic potentials in terms of the existence of a solution to a scalar Riemann--Hilbert problem, and we derive a similar condition on the spectrum for the temporal periodicity for an evolution under the KdV equation.