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

Athmane Bakhta; Virginie Ehrlacher; David Gontier

doi:10.1051/cocv/2019031

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

Athmane Bakhta
, Virginie Ehrlacher
, David Gontier

Université Paris Est, ENPC LIGM, IMAGINE
Université Paris Dauphine

Research output: Contribution to journal › Article › peer-review

Abstract

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).

Original language	English
Article number	59
Journal	ESAIM - Control, Optimisation and Calculus of Variations
Volume	26
DOIs	https://doi.org/10.1051/cocv/2019031
Publication status	Published - 1 Jan 2020
Externally published	Yes

Keywords

Band structure
Hill's operator
Inverse spectral theory
Optimisation
Periodic Schrödinger operator

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10.1051/cocv/2019031

Cite this

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abstract = "This paper concerns an inverse band structure problem for one dimensional periodic Schr{\"o}dinger 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).",

keywords = "Band structure, Hill's operator, Inverse spectral theory, Optimisation, Periodic Schr{\"o}dinger operator",

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AU - Ehrlacher, Virginie

AU - Gontier, David

N1 - Publisher Copyright: © EDP Sciences, SMAI 2020.

PY - 2020/1/1

Y1 - 2020/1/1

N2 - 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).

AB - 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).

KW - Band structure

KW - Hill's operator

KW - Inverse spectral theory

KW - Optimisation

KW - Periodic Schrödinger operator

UR - https://www.scopus.com/pages/publications/85092300850

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DO - 10.1051/cocv/2019031

M3 - Article

AN - SCOPUS:85092300850

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VL - 26

JO - ESAIM - Control, Optimisation and Calculus of Variations

JF - ESAIM - Control, Optimisation and Calculus of Variations

M1 - 59

ER -

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

Abstract

Keywords

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