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6 results on '"35R30, 35Q41"'

1. H\'older stably determining the time-dependent electromagnetic potential of the Schr\'odinger equation

Author

Kian, Yavar and Soccorsi, Eric

Subjects
Mathematics - Analysis of PDEs, 35R30, 35Q41
Abstract

We consider the inverse problem of determining the time and space dependent electromagnetic potential of the Schr\"odinger equation in a bounded domain of $\mathbb R^n$, $n\geq 2$, by boundary observation of the solution over the entire time span. Assuming that the divergence of the magnetic potential is fixed, we prove that the electric potential and the magnetic potential can be H\"older stably retrieved from these data, whereas stability estimates for inverse time-dependent coefficients problems of evolution partial differential equations are usually of logarithmic type.

Published
2017

2. An inverse problem for the magnetic Schr\'odinger equation in infinite cylindrical domains

Author

Bellassoued, Mourad, Kian, Yavar, and Soccorsi, Eric

Subjects
Mathematics - Analysis of PDEs, 35R30, 35Q41
Abstract

We study the inverse problem of determining the magnetic field and the electric potential entering the Schr\"odinger equation in an infinite 3D cylindrical domain, by Dirichlet-to-Neumann map. The cylindrical domain we consider is a closed waveguide in the sense that the cross section is a bounded domain of the plane. We prove that the knowledge of the Dirichlet-to-Neumann map determines uniquely, and even H\"older-stably, the magnetic field induced by the magnetic potential and the electric potential. Moreover, if the maximal strength of both the magnetic field and the electric potential, is attained in a fixed bounded subset of the domain, we extend the above results by taking finitely extended boundary observations of the solution, only.

Published
2016

3. An inverse stability result for non compactly supported potentials by one arbitrary lateral Neumann observation

Author

Bellassoued, Mourad, Kian, Yavar, and Soccorsi, Eric

Subjects
Mathematics - Analysis of PDEs, 35R30, 35Q41
Abstract

In this paper we investigate the inverse problem of determining the time independent scalar potential of the dynamic Schr\"odinger equation in an infinite cylindrical domain, from partial measurement of the solution on the boundary. Namely, if the potential is known in a neighborhood of the boundary of the spatial domain, we prove that it can be logarithmic stably determined in the whole waveguide from a single observation of the solution on any arbitrary strip-shaped subset of the boundary.

Published
2015

4. An inverse stability result for non-compactly supported potentials by one arbitrary lateral Neumann observation

Author

Mourad Bellassoued, Yavar Kian, Eric Soccorsi, Fédération de recherche Denis Poisson (FDP), Université d'Orléans (UO)-Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS), Université de Carthage - University of Carthage, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO), and Université d'Orléans (UO)-Université de Tours-Centre National de la Recherche Scientifique (CNRS)

Subjects
Infinite cylindrical waveguide, Logarithm, Inverse Problems, Boundary (topology), Inverse, Schrödinger equation, Scalar potential, Carleman estimate, 01 natural sciences, Domain (mathematical analysis), 35R30, 35Q41, symbols.namesake, Stability estimate, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Waveguide (acoustics), 0101 mathematics, Mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Inverse problem, 16. Peace & justice, 010101 applied mathematics, symbols, Analysis
Abstract

International audience; In this paper we investigate the inverse problem of determining the time independent scalar potential of the dynamic Schrödinger equation in an infinite cylindrical domain, from partial measurement of the solution on the boundary. Namely, if the potential is known in a neighborhood of the boundary of the spatial domain, we prove that it can be logarithmic stably determined in the whole waveguide from a single observation of the solution on any arbitrary strip-shaped subset of the boundary.

Published
2016
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5. An inverse problem for the magnetic Schrödinger equation in infinite cylindrical domains

Author

Mourad Bellassoued, Yavar Kian, Eric Soccorsi, Laboratoire de Modélisation Mathématique et Numérique dans les Sciences de l'Ingénieur [Tunis] (LR-LAMSIN-ENIT), Ecole Nationale d'Ingénieurs de Tunis (ENIT), Université de Tunis El Manar (UTM)-Université de Tunis El Manar (UTM), Université de Tunis El Manar (UTM), CPT - E8 Dynamique quantique et analyse spectrale, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
magnetic Schrödinger equation, Plane (geometry), General Mathematics, 010102 general mathematics, Mathematical analysis, Dirichlet-to-Neumann map, Mathematics::Spectral Theory, Inverse problem, infinite cylindrical domain, 01 natural sciences, Domain (mathematical analysis), Magnetic field, Schrödinger equation, 010101 applied mathematics, symbols.namesake, 35R30, 35Q41, Mathematics - Analysis of PDEs, Bounded function, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Electric potential, Magnetic potential, 0101 mathematics, Mathematics
Abstract

International audience; We study the inverse problem of determining the magnetic field and the electric potential entering the Schrödinger equation in an infinite 3D cylindrical domain, by Dirichlet-to-Neumann map. The cylindrical domain we consider is a closed waveguide in the sense that the cross section is a bounded domain of the plane. We prove that the knowledge of the Dirichlet-to-Neumann map determines uniquely, and even Hölder-stably, the magnetic field induced by the magnetic potential and the electric potential. Moreover, if the maximal strength of both the magnetic field and the electric potential, is attained in a fixed bounded subset of the domain, we extend the above results by taking finitely extended boundary observations of the solution, only.

Published
2018
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6. H\'older stably determining the time-dependent electromagnetic potential of the Schr\'odinger equation

Author

Eric Soccorsi, Yavar Kian, CPT - E8 Dynamique quantique et analyse spectrale, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)

Subjects
Boundary (topology), Schrödinger equation, 01 natural sciences, Domain (mathematical analysis), symbols.namesake, 35R30, 35Q41, Mathematics - Analysis of PDEs, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematics, Partial differential equation, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Inverse problem, time-dependent electromagnetic potential, 010101 applied mathematics, Computational Mathematics, Bounded function, symbols, stability estimate, Electric potential, Magnetic potential, Analysis
Abstract

International audience; We consider the inverse problem of determining the time and space dependent electromagnetic potential of the Schrödinger equation in a bounded domain of R n , n 2, by boundary observation of the solution over the entire time span. Assuming that the divergence of the magnetic potential is fixed, we prove that the electric potential and the magnetic potential can be Hölder stably retrieved from these data, whereas stability estimates for inverse time-dependent coefficients problems of evolution partial differential equations are usually of logarithmic type.

Published
2017
Full Text
View/download PDF

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