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

Athmane Bakhta (1) , Virginie Ehrlacher (2) , David Gontier (3)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 MATHERIALS - MATHematics for MatERIALS
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

This paper concerns an inverse band structure problem for one dimensional periodic Schrödinger 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.

Mots clés: Inverse Hill; Optimisation; Inverse band structure; Periodic Schrödinger operator

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]; Théorie spectrale [math.SP]; Optimisation et contrôle [math.OC]

Athmane Bakhta, Virginie Ehrlacher, David Gontier. Numerical reconstruction of the first band(s) in an inverse Hill's problem. ESAIM: Control, Optimisation and Calculus of Variations, 2020, ⟨10.1051/cocv/2019031⟩. ⟨hal-01591133⟩