Inverse spectral theory for uniformly open gaps in a weighted Sturm–Liouville problem

Motivated by a PDE existence problem, we study the inverse problem for a weighted Sturm–Liouville operator LsLs associated with the eigenvalue problem y″+λs(x)y=0, where s   is a real-valued, periodic, even function that is bounded from below by a positive constant and belongs to the L2L2-based Sobolev space Hr[0,1]Hr[0,1], r≥1r≥1. Choosing gap lengths and gap midpoints as coordinates, we define a spectral map GG, that assigns to a coefficient s   the structure of the spectrum of LsLs. We find that GG is a real-analytic isomorphism locally around s=1s=1, which, in particular, implies the existence of coefficients s∈Hr[0,1]s∈Hr[0,1], r<3/2r<3/2, whose spectrum features band structure with all gaps uniformly open around the gap midpoints. This result paves the way for the construction of so-called breathers in nonlinear wave equations with such coefficients s  . Apart from the novelty of treating the inverse spectral problem for the full Banach scale Hr[0,1]Hr[0,1], r≥1r≥1, the local nature of our result allows more concise and transparent proofs. In particular, instead of using any preliminary transformations, we treat the weighted problem directly by adapting techniques used for Schrödinger operators with distribution potentials.