An inverse problem for the magnetic Schrödinger operator on a half space with partial data

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November 2014, 8(4): 1169-1189. doi: 10.3934/ipi.2014.8.1169

An inverse problem for the magnetic Schrödinger operator on a half space with partial data

Valter Pohjola ^1,

1.	Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 (Gustaf Hallstromin katu 2b) FI-00014, Finland

Received April 2013 Revised November 2013 Published November 2014

Abstract
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In this paper we prove uniqueness for an inverse boundary value problem for the magnetic Schrödinger equation in a half space, with partial data. We prove that the curl of the magnetic potential $A$, when $A\in W_{comp}^{1,\infty}(\overline{\mathbb{R}_{-}^3},\mathbb{R}^3)$, and the electric pontetial $q \in L_{comp}^{\infty}(\overline{\mathbb{R}_{-}^3},\mathbb{C})$ are uniquely determined by the knowledge of the Dirichlet-to-Neumann map on parts of the boundary of the half space.

Keywords: magnetic Schrödinger operator, Inverse boundary value problem, half space, uniqueness, partial data..

Mathematics Subject Classification: Primary: 35R3.

Citation: Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems and Imaging, 2014, 8 (4) : 1169-1189. doi: 10.3934/ipi.2014.8.1169

References:

[1]	S. Arridge, Optical tomography in medical imaging, Inverse Problems, 15 (1999), R41-R93. doi: 10.1088/0266-5611/15/2/022.
[2]	M. Cheney and D. Isaacson, Inverse problems for a perturbed dissipative half-space, Inverse Problems, 11 (1995), 865-888. doi: 10.1088/0266-5611/11/4/015.
[3]	M. Choulli, Une Introduction Aux Problèmes Inverses Elliptiques et Paraboliques, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-02460-3.
[4]	F. J. Chung, A partial data result for the magnetic Schrödinger inverse problem, Anal. PDE, 7 (2014), 117-157. arXiv:1111.6658. doi: 10.2140/apde.2014.7.117.
[5]	D. Dos Santos Ferreira, C. E. Kenig, J. Sjöstrand and G. Uhlmann, Determining a magnetic Schrödinger operator from partial Cauchy data, Communications in Mathematical Physics, 271 (2007), 467-488. doi: 10.1007/s00220-006-0151-9.
[6]	G. Eskin and J. Ralston, Inverse scattering problem for the Schrödinger equation with magentic potential at a fixed energy, Communications in Mathematicala Physics, 173 (1995), 199-224. doi: 10.1007/BF02100187.
[7]	L. Hörmander, An Introduction to Complex Analysis in Several Variables, North-Holland, Amsterdam, 1990.
[8]	V. Isakov, On uniqueness in the inverse conductivity problem with local data, Inverse Problems and Imaging, 1 (2007), 95-105. doi: 10.3934/ipi.2007.1.95.
[9]	K. Knudsen and M. Salo, Determining nonsmooth first order terms from partial boundary measurements, Inverse Problems and Imaging, 1 (2007), 349-369. doi: 10.3934/ipi.2007.1.349.
[10]	R. Kress, Linear Integral Equations, Springer, Berlin, 1989. doi: 10.1007/978-3-642-97146-4.
[11]	K. Krupchyk, M. Lassas and G. Uhlmann, Inverse problems with partial data for a magnetic Schrödinger operator in an infinite slab and on a bounded domain, Communications in Mathematical Physics, 312 (2012), 87-126. doi: 10.1007/s00220-012-1431-1.
[12]	K. Krupchyk and G. Uhlmann, Uniqueness in an inverse boundary problem for a magnetic Schrödinger operator with a bounded magnetic potential, Comm. Math. Phys., 327 (2014), 993-1009. doi: 10.1007/s00220-014-1942-z.
[13]	M. Lassas, M. Cheney and G. Uhlmann, Uniqueness for a wave propagation inverse problem in a half-space, Inverse Problems, 14 (1998), 679-684. doi: 10.1088/0266-5611/14/3/017.
[14]	R. Leis, Initial Boundary Values Problems in Mathematical Physics, John Wiley and Sons Ltd., Stuttgart, 1986. doi: 10.1007/978-3-663-10649-4.
[15]	X. Li, Inverse boundary value problems with partial data in unbounded domains, Inverse Problems, 28 (2012), 1-23. doi: 10.1088/0266-5611/28/8/085003.
[16]	X. Li and G. Uhlmann, Inverse problems with partial data in a slab, Inverse Problems and Imaging, 4 (2010), 449-462. doi: 10.3934/ipi.2010.4.449.
[17]	G. Nakamura, Z. Sun and G. Uhlmann, Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field, Mathematische Annalen, 303 (1995), 377-388. doi: 10.1007/BF01460996.
[18]	A. Panchenko, An inverse problem for the magnetic schrödinger equation and quasi-exponential solutions of nonsmooth partial differential equations, Inverse Problems, 18 (2002), 1421-1434. doi: 10.1088/0266-5611/18/5/314.
[19]	V. Pohjola, An Inverse Boundary Value Problem for the Magnetic Schrödinger Operator on a Half Space, preprint, 2012, arXiv:1209.0982.
[20]	M. Salo, Inverse problems for nonsmooth first order perturbations of the Laplacian, Ann. Acad. Sci. Fenn. Math. Diss., 139, 2004, 67 pp.
[21]	M. Salo, Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field, Communications in Partial Differential Equations, 31 (2006), 1639-1666. doi: 10.1080/03605300500530420.
[22]	Z. Sun, An inverse boundary problem for Schrödinger operators with vector potentials, Trans. Amer. Math.Soc., 338 (1993), 953-969. doi: 10.2307/2154438.
[23]	C. Tolmasky, Exponentially growing solutions for nonsmooth first-order perturbations of the laplacian, SIAM J. Math. Anal., 29 (1998), 116-133. doi: 10.1137/S0036141096301038.
[24]	G. Uhlmann, electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 123011. doi: 10.1088/0266-5611/25/12/123011.

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References:

[1]	S. Arridge, Optical tomography in medical imaging, Inverse Problems, 15 (1999), R41-R93. doi: 10.1088/0266-5611/15/2/022.
[2]	M. Cheney and D. Isaacson, Inverse problems for a perturbed dissipative half-space, Inverse Problems, 11 (1995), 865-888. doi: 10.1088/0266-5611/11/4/015.
[3]	M. Choulli, Une Introduction Aux Problèmes Inverses Elliptiques et Paraboliques, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-02460-3.
[4]	F. J. Chung, A partial data result for the magnetic Schrödinger inverse problem, Anal. PDE, 7 (2014), 117-157. arXiv:1111.6658. doi: 10.2140/apde.2014.7.117.
[5]	D. Dos Santos Ferreira, C. E. Kenig, J. Sjöstrand and G. Uhlmann, Determining a magnetic Schrödinger operator from partial Cauchy data, Communications in Mathematical Physics, 271 (2007), 467-488. doi: 10.1007/s00220-006-0151-9.
[6]	G. Eskin and J. Ralston, Inverse scattering problem for the Schrödinger equation with magentic potential at a fixed energy, Communications in Mathematicala Physics, 173 (1995), 199-224. doi: 10.1007/BF02100187.
[7]	L. Hörmander, An Introduction to Complex Analysis in Several Variables, North-Holland, Amsterdam, 1990.
[8]	V. Isakov, On uniqueness in the inverse conductivity problem with local data, Inverse Problems and Imaging, 1 (2007), 95-105. doi: 10.3934/ipi.2007.1.95.
[9]	K. Knudsen and M. Salo, Determining nonsmooth first order terms from partial boundary measurements, Inverse Problems and Imaging, 1 (2007), 349-369. doi: 10.3934/ipi.2007.1.349.
[10]	R. Kress, Linear Integral Equations, Springer, Berlin, 1989. doi: 10.1007/978-3-642-97146-4.
[11]	K. Krupchyk, M. Lassas and G. Uhlmann, Inverse problems with partial data for a magnetic Schrödinger operator in an infinite slab and on a bounded domain, Communications in Mathematical Physics, 312 (2012), 87-126. doi: 10.1007/s00220-012-1431-1.
[12]	K. Krupchyk and G. Uhlmann, Uniqueness in an inverse boundary problem for a magnetic Schrödinger operator with a bounded magnetic potential, Comm. Math. Phys., 327 (2014), 993-1009. doi: 10.1007/s00220-014-1942-z.
[13]	M. Lassas, M. Cheney and G. Uhlmann, Uniqueness for a wave propagation inverse problem in a half-space, Inverse Problems, 14 (1998), 679-684. doi: 10.1088/0266-5611/14/3/017.
[14]	R. Leis, Initial Boundary Values Problems in Mathematical Physics, John Wiley and Sons Ltd., Stuttgart, 1986. doi: 10.1007/978-3-663-10649-4.
[15]	X. Li, Inverse boundary value problems with partial data in unbounded domains, Inverse Problems, 28 (2012), 1-23. doi: 10.1088/0266-5611/28/8/085003.
[16]	X. Li and G. Uhlmann, Inverse problems with partial data in a slab, Inverse Problems and Imaging, 4 (2010), 449-462. doi: 10.3934/ipi.2010.4.449.
[17]	G. Nakamura, Z. Sun and G. Uhlmann, Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field, Mathematische Annalen, 303 (1995), 377-388. doi: 10.1007/BF01460996.
[18]	A. Panchenko, An inverse problem for the magnetic schrödinger equation and quasi-exponential solutions of nonsmooth partial differential equations, Inverse Problems, 18 (2002), 1421-1434. doi: 10.1088/0266-5611/18/5/314.
[19]	V. Pohjola, An Inverse Boundary Value Problem for the Magnetic Schrödinger Operator on a Half Space, preprint, 2012, arXiv:1209.0982.
[20]	M. Salo, Inverse problems for nonsmooth first order perturbations of the Laplacian, Ann. Acad. Sci. Fenn. Math. Diss., 139, 2004, 67 pp.
[21]	M. Salo, Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field, Communications in Partial Differential Equations, 31 (2006), 1639-1666. doi: 10.1080/03605300500530420.
[22]	Z. Sun, An inverse boundary problem for Schrödinger operators with vector potentials, Trans. Amer. Math.Soc., 338 (1993), 953-969. doi: 10.2307/2154438.
[23]	C. Tolmasky, Exponentially growing solutions for nonsmooth first-order perturbations of the laplacian, SIAM J. Math. Anal., 29 (1998), 116-133. doi: 10.1137/S0036141096301038.
[24]	G. Uhlmann, electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 123011. doi: 10.1088/0266-5611/25/12/123011.

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