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

Bakhta , Athmane; Ehrlacher, Virginie; Gontier, David (2020), Numerical reconstruction of the first band(s) in an inverse Hill's problem, ESAIM. Control, Optimisation and Calculus of Variations, 26, 59. 10.1051/cocv/2019031

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Type
Article accepté pour publication ou publié
Date
2020
Journal name
ESAIM. Control, Optimisation and Calculus of Variations
Volume
26
Number
59
Publisher
EDP Sciences
Publication identifier
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Author(s)
Bakhta , Athmane
École des Ponts ParisTech [ENPC]
Ehrlacher, Virginie
École des Ponts ParisTech [ENPC]
Gontier, David
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
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.
Subjects / Keywords
Inverse Hill; optimisation; inverse band structure; periodic Schrödinger operator