Stability estimate in the inverse scattering for a single quantum particle in an external short-range potential

Mourad Bellassoued; Luc Robbiano

doi:10.1016/j.crma.2019.09.009

Partial differential equations

Stability estimate in the inverse scattering for a single quantum particle in an external short-range potential
[Estimation de stabilité en diffusion inverse pour une particule quantique en présence d'un potentiel à courte portée]

Présenté par : the Editorial Board

Mourad Bellassoued ¹ ; Luc Robbiano ²

¹ Université de Tunis El Manar, École nationale d'ingénieurs de Tunis, LAMSIN, B.P. 37, 1002 Tunis, Tunisia
² Laboratoire de mathématiques, Université de Versailles – Saint-Quentin-en-Yvelines, 78035 Versailles, France

Comptes Rendus. Mathématique, Volume 357 (2019) no. 10, pp. 784-798.

Résumé
Abstract

Dans cet article, nous considérons le problème de la diffusion inverse pour l'opérateur de Schrödinger avec un potentiel électrique à courte portée. Nous prouvons en dimension $n \geq 2$ que la connaissance de l'opérateur de diffusion détermine le potentiel électrique et nous établissons une estimation de stabilité de type Hölder pour la détermination du potentiel électrique à courte portée.

In this paper we consider the inverse scattering problem for the Schrödinger operator with short-range electric potential. We prove in dimension $n \geq 2$ that the knowledge of the scattering operator determines the electric potential and we establish Hölder-type stability in determining the short range electric potential.

Reçu le : 2018-09-04
Accepté le : 2019-09-16
Publié le : 2019-10-01

DOI : 10.1016/j.crma.2019.09.009

Affiliations des auteurs :

Mourad Bellassoued ¹ ; Luc Robbiano ²

¹ Université de Tunis El Manar, École nationale d'ingénieurs de Tunis, LAMSIN, B.P. 37, 1002 Tunis, Tunisia
² Laboratoire de mathématiques, Université de Versailles – Saint-Quentin-en-Yvelines, 78035 Versailles, France

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Mourad Bellassoued; Luc Robbiano. Stability estimate in the inverse scattering for a single quantum particle in an external short-range potential. Comptes Rendus. Mathématique, Volume 357 (2019) no. 10, pp. 784-798. doi : 10.1016/j.crma.2019.09.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.09.009/

Bibliographie
Cité par

[1] S. Arians Geometric approach to inverse scattering for the Schrödinger equation with magnetic and electric potentials, J. Math. Phys., Volume 38 (1997) no. 6, pp. 2761-2773

[2] M. Bellassoued; D. Jellali; M. Yamamoto Stability estimate for the hyperbolic inverse boundary value problem by local Dirichlet-to-Neumann map, J. Math. Anal. Appl., Volume 343 (2008) no. 2, pp. 1036-1046

[3] A.L. Bukhgeim Recovering the potential from Cauchy data in two dimensions, J. Inverse Ill-Posed Probl., Volume 16 (2008), pp. 19-33

[4] V. Enss Propagation properties of quantum scattering states, J. Funct. Anal., Volume 52 (1983), pp. 219-251

[5] V. Enss Quantum scattering with long-range magnetic fields (M. Demuth; B. Gramsch; B.-W. Schulze, eds.), Operator Calculus and Spectral Theory, Operator Theory: Advances and Applications, vol. 57, Birkhäuser, Basel, Switzerland, 1992, pp. 61-70

[6] V. Enss; R. Weder The geometrical approach to multidimensional inverse scattering, J. Math. Phys., Volume 36 (1995), pp. 3902-3921

[7] G. Eskin; J. Ralston Inverse scattering problems for the Schrödinger equation with magnetic potential at a fixed energy, Commun. Math. Phys., Volume 173 (1995), pp. 199-224

[8] P. Grinevich; R.G. Novikov Transparent potentials at fixed energy in dimension two. Fixed-energy dispersion relations for the fast decaying potentials, Commun. Math. Phys., Volume 174 (1995), pp. 409-446

[9] C. Guillarmou; M. Salo; L. Tzou Inverse scattering at fixed energy on surfaces with Euclidean ends, Commun. Math. Phys., Volume 303 (2011), pp. 761-784

[10] L. Hörmander The Analysis of Linear Partial Differential Operators, Vol. I, II, Springer-Verlag, Berlin, 1983

[11] M. Joshi Explicitly recovering asymptotics of short range potentials, Commun. Partial Differ. Equ., Volume 25 (2000), pp. 1907-1923

[12] M. Joshi; A. Sá Barreto Determining asymptotics of magnetic fields from fixed energy scattering data, Asymptot. Anal., Volume 21 (1999), pp. 61-70

[13] W. Jung Geometrical approach to inverse scattering for the Dirac equation, J. Math. Phys., Volume 38 (1997), pp. 39-48

[14] R.B. Melrose Geometric Scattering Theory, Cambridge University Press, 1995

[15] A. Nachman Reconstructions from boundary measurements, Ann. of Math. (2), Volume 128 (1988), pp. 531-576

[16] R.G. Newton Construction of potentials from the phase shifts at fixed energy, J. Math. Phys., Volume 3 (1962), pp. 75-82

[17] F. Nicoleau A Stationary Approach to Inverse Scattering for Schrödinger Operators with First Order Perturbations, Université de Nantes, France, 1996 (Rapport de recherche 96/4-2)

[18] F. Nicoleau Inverse Scattering for Schrödinger Operators with First Order Perturbations: The Long-Range Case, Université de Nantes, France, 1996 (Rapport de recherche 96/5-2)

[19] F. Nicoleau A stationary approach to inverse scattering for Schröinger operators with first order perturbation, Commun. Partial Differ. Equ., Volume 22 (1997), pp. 527-553

[20] F. Nicoleau An inverse scattering problem with the Aharonov-Bohm effect, J. Math. Phys., Volume 41 (2000), p. 5223

[21] R.G. Novikov A multidimensional inverse spectral problem for the equation $Δ ψ + (v (x) - Eu (x)) ψ = 0$ , Funct. Anal. Appl., Volume 22 (1988), pp. 263-272

[22] R.G. Novikov The inverse scattering problem at fixed energy for the three-dimensional Schrödinger equation with an exponentially decreasing potential, Commun. Math. Phys., Volume 161 (1994), pp. 569-595

[23] R.G. Novikov; G.M. Khenkin The $\bar{\partial}$ -equation in the multidimensional inverse scattering problem, Russ. Math. Surv., Volume 42 (1987), pp. 109-180

[24] A.G. Ramm Recovery of the potential from fixed energy scattering data, Inverse Probl., Volume 4 (1988), pp. 877-886

[25] M. Reed; B. Simon Methods of Modern Mathematical Physics, Vol. III. Scattering Theory, Academic Press, New York, 1979

[26] P. Sabatier Asymptotic properties of the potentials in the inverse-scattering problem at fixed energy, J. Math. Phys., Volume 7 (1966), pp. 1515-1531

[27] Y. Saito Some properties of the scattering amplitude and the inverse scattering problem, Osaka J. Math., Volume 19 (1982), pp. 527-547

[28] J. Sylvester; G. Uhlmann A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2), Volume 125 (1987), pp. 153-169

[29] G. Uhlmann Inverse boundary value problems and applications, Astérisque, Volume 207 (1992), pp. 153-211

[30] G. Uhlmann; A. Vasy Fixed energy inverse problem for exponentially decreasing potentials, Methods Appl. Anal., Volume 9 (2002), pp. 239-248

[31] R. Weder Multidimensional inverse scattering in an electric field, J. Funct. Anal., Volume 139 (1996) no. 2, pp. 441-465

[32] R. Weder The Aharonov–Bohm effect and time-dependent inverse scattering theory, Inverse Probl., Volume 18 (2002) no. 4, pp. 1041-1056

[33] R. Weder Completeness of averaged scattering solutions, Commun. Partial Differ. Equ., Volume 32 (2007), pp. 675-691

[34] R. Weder; D. Yafaev On inverse scattering at a fixed energy for potentials with a regular behaviour at infinity, Inverse Probl., Volume 21 (2005), pp. 1937-1952

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