Numerical reconstruction of the first band(s) in an inverse Hill's problem

TL;DR

This paper develops a numerical approach to reconstruct potentials in one-dimensional periodic Schrödinger operators to match target band structures, proving well-posedness for singular potentials and proposing algorithms for practical computation.

Contribution

It introduces a well-posed formulation for inverse band problems with singular potentials and offers new algorithms for numerical reconstruction.

Findings

Proved well-posedness for inverse band problems with Borel measure potentials

Developed algorithms for numerical potential reconstruction

Demonstrated effectiveness through computational experiments

Abstract

This paper concerns an inverse band structure problem for one dimensional periodic Schr\"odinger operators (Hill's operators). Our goal is to find a potential for the Hill's operator in order to reproduce as best as possible some given target bands, which may not be realisable. We recast the problem as an optimisation problem, and prove that this problem is well-posed when considering singular potentials (Borel measures). We then propose different algorithms to tackle the problem numerically.