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

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 & 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). doi: 10.1088/0266-5611/15/2/022. Google Scholar
[2]	M. Cheney and D. Isaacson, Inverse problems for a perturbed dissipative half-space,, Inverse Problems, 11 (1995), 865. doi: 10.1088/0266-5611/11/4/015. Google Scholar
[3]	M. Choulli, Une Introduction Aux Problèmes Inverses Elliptiques et Paraboliques,, Springer-Verlag, (2009). doi: 10.1007/978-3-642-02460-3. Google Scholar
[4]	F. J. Chung, A partial data result for the magnetic Schrödinger inverse problem,, Anal. PDE, 7 (2014), 117. doi: 10.2140/apde.2014.7.117. Google Scholar
[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. doi: 10.1007/s00220-006-0151-9. Google Scholar*
[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. doi: 10.1007/BF02100187. Google Scholar*
[7]	L. Hörmander, An Introduction to Complex Analysis in Several Variables,, North-Holland, (1990). Google Scholar
[8]	V. Isakov, On uniqueness in the inverse conductivity problem with local data,, Inverse Problems and Imaging, 1 (2007), 95. doi: 10.3934/ipi.2007.1.95. Google Scholar
[9]	K. Knudsen and M. Salo, Determining nonsmooth first order terms from partial boundary measurements,, Inverse Problems and Imaging, 1 (2007), 349. doi: 10.3934/ipi.2007.1.349. Google Scholar
[10]	R. Kress, Linear Integral Equations,, Springer, (1989). doi: 10.1007/978-3-642-97146-4. Google Scholar
[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. doi: 10.1007/s00220-012-1431-1. Google Scholar*
[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. doi: 10.1007/s00220-014-1942-z. Google Scholar*
[13]	M. Lassas, M. Cheney and G. Uhlmann, Uniqueness for a wave propagation inverse problem in a half-space,, Inverse Problems, 14 (1998), 679. doi: 10.1088/0266-5611/14/3/017. Google Scholar
[14]	R. Leis, Initial Boundary Values Problems in Mathematical Physics,, John Wiley and Sons Ltd., (1986). doi: 10.1007/978-3-663-10649-4. Google Scholar
[15]	X. Li, Inverse boundary value problems with partial data in unbounded domains,, Inverse Problems, 28 (2012), 1. doi: 10.1088/0266-5611/28/8/085003. Google Scholar
[16]	X. Li and G. Uhlmann, Inverse problems with partial data in a slab,, Inverse Problems and Imaging, 4 (2010), 449. doi: 10.3934/ipi.2010.4.449. Google Scholar
[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. doi: 10.1007/BF01460996. Google Scholar*
[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. doi: 10.1088/0266-5611/18/5/314. Google Scholar
[19]	V. Pohjola, An Inverse Boundary Value Problem for the Magnetic Schrödinger Operator on a Half Space,, preprint, (2012). Google Scholar
[20]	M. Salo, Inverse problems for nonsmooth first order perturbations of the Laplacian,, Ann. Acad. Sci. Fenn. Math. Diss., 139* (2004). Google Scholar*
[21]	M. Salo, Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field,, Communications in Partial Differential Equations, 31 (2006), 1639. doi: 10.1080/03605300500530420. Google Scholar
[22]	Z. Sun, An inverse boundary problem for Schrödinger operators with vector potentials,, Trans. Amer. Math.Soc., 338* (1993), 953. doi: 10.2307/2154438. Google Scholar*
[23]	C. Tolmasky, Exponentially growing solutions for nonsmooth first-order perturbations of the laplacian,, SIAM J. Math. Anal., 29 (1998), 116. doi: 10.1137/S0036141096301038. Google Scholar
[24]	G. Uhlmann, electrical impedance tomography and Calderón's problem,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/12/123011. Google Scholar

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References:

[1]	S. Arridge, Optical tomography in medical imaging,, Inverse Problems, 15 (1999). doi: 10.1088/0266-5611/15/2/022. Google Scholar
[2]	M. Cheney and D. Isaacson, Inverse problems for a perturbed dissipative half-space,, Inverse Problems, 11 (1995), 865. doi: 10.1088/0266-5611/11/4/015. Google Scholar
[3]	M. Choulli, Une Introduction Aux Problèmes Inverses Elliptiques et Paraboliques,, Springer-Verlag, (2009). doi: 10.1007/978-3-642-02460-3. Google Scholar
[4]	F. J. Chung, A partial data result for the magnetic Schrödinger inverse problem,, Anal. PDE, 7 (2014), 117. doi: 10.2140/apde.2014.7.117. Google Scholar
[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. doi: 10.1007/s00220-006-0151-9. Google Scholar*
[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. doi: 10.1007/BF02100187. Google Scholar*
[7]	L. Hörmander, An Introduction to Complex Analysis in Several Variables,, North-Holland, (1990). Google Scholar
[8]	V. Isakov, On uniqueness in the inverse conductivity problem with local data,, Inverse Problems and Imaging, 1 (2007), 95. doi: 10.3934/ipi.2007.1.95. Google Scholar
[9]	K. Knudsen and M. Salo, Determining nonsmooth first order terms from partial boundary measurements,, Inverse Problems and Imaging, 1 (2007), 349. doi: 10.3934/ipi.2007.1.349. Google Scholar
[10]	R. Kress, Linear Integral Equations,, Springer, (1989). doi: 10.1007/978-3-642-97146-4. Google Scholar
[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. doi: 10.1007/s00220-012-1431-1. Google Scholar*
[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. doi: 10.1007/s00220-014-1942-z. Google Scholar*
[13]	M. Lassas, M. Cheney and G. Uhlmann, Uniqueness for a wave propagation inverse problem in a half-space,, Inverse Problems, 14 (1998), 679. doi: 10.1088/0266-5611/14/3/017. Google Scholar
[14]	R. Leis, Initial Boundary Values Problems in Mathematical Physics,, John Wiley and Sons Ltd., (1986). doi: 10.1007/978-3-663-10649-4. Google Scholar
[15]	X. Li, Inverse boundary value problems with partial data in unbounded domains,, Inverse Problems, 28 (2012), 1. doi: 10.1088/0266-5611/28/8/085003. Google Scholar
[16]	X. Li and G. Uhlmann, Inverse problems with partial data in a slab,, Inverse Problems and Imaging, 4 (2010), 449. doi: 10.3934/ipi.2010.4.449. Google Scholar
[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. doi: 10.1007/BF01460996. Google Scholar*
[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. doi: 10.1088/0266-5611/18/5/314. Google Scholar
[19]	V. Pohjola, An Inverse Boundary Value Problem for the Magnetic Schrödinger Operator on a Half Space,, preprint, (2012). Google Scholar
[20]	M. Salo, Inverse problems for nonsmooth first order perturbations of the Laplacian,, Ann. Acad. Sci. Fenn. Math. Diss., 139* (2004). Google Scholar*
[21]	M. Salo, Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field,, Communications in Partial Differential Equations, 31 (2006), 1639. doi: 10.1080/03605300500530420. Google Scholar
[22]	Z. Sun, An inverse boundary problem for Schrödinger operators with vector potentials,, Trans. Amer. Math.Soc., 338* (1993), 953. doi: 10.2307/2154438. Google Scholar*
[23]	C. Tolmasky, Exponentially growing solutions for nonsmooth first-order perturbations of the laplacian,, SIAM J. Math. Anal., 29 (1998), 116. doi: 10.1137/S0036141096301038. Google Scholar
[24]	G. Uhlmann, electrical impedance tomography and Calderón's problem,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/12/123011. Google Scholar

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