Hill's potentials in weighted Sobolev spaces and their spectral gaps

TL;DR

This paper presents a concise proof linking the spectral gap lengths of Hill's equation potentials to their regularity in weighted Sobolev spaces, extending known results to complex potentials and utilizing the implicit function theorem.

Contribution

It provides a new, streamlined proof of the relationship between spectral gaps and potential regularity, including extensions to complex potentials and recovery of classical results.

Findings

Spectral gap lengths characterize potential regularity in weighted Sobolev spaces.

Extension of results to complex potentials.

Recovery of Trubowitz's results on analytic potentials.

Abstract

We describe a new, short proof of some facts relating the gap lengths of the spectrum of a potential of Hill's equation to its regularity. For example, a real potential is in a weighted Gevrey-Sobolev space if and only if its gap lengths belong to a similarly weighted sequence space. An extension of this result to complex potentials is proven as well. We also recover Trubowitz results about analytic potentials. The proof essentially employs the implicit function theorem.