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

We formulate the inverse spectral theory of infinite gap Hill’s operators with bounded periodic potentials 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 potentials evolve according to the Korteweg–de Vries Equation (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.