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Inverse Problems With Partial Data for a Magnetic Schrödinger Operator in an Infinite Slab and on a Bounded Domain

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Abstract

In this paper we study inverse boundary value problems with partial data for the magnetic Schrödinger operator. In the case of an infinite slab in \({\mathbb{R}^n}\) , n ≥ 3, we establish that the magnetic field and the electric potential can be determined uniquely, when the Dirichlet and Neumann data are given either on the different boundary hyperplanes of the slab or on the same hyperplane. This is a generalization of the results of Li and Uhlmann (Inverse Probl Imaging 4(3):449–462, 2010), obtained for the Schrödinger operator without magnetic potentials.

In the case of a bounded domain in \({\mathbb{R}^n}\) , n ≥ 3, extending the results of Ammari and Uhlmann (Indiana Univ Math J 53(1):169–183, 2004), we show the unique determination of the magnetic field and electric potential from the Dirichlet and Neumann data, given on two arbitrary open subsets of the boundary, provided that the magnetic and electric potentials are known in a neighborhood of the boundary. Generalizing the results of Isakov (Inverse Probl Imaging 1(1):95–105, 2007), we also obtain uniqueness results for the magnetic Schrödinger operator, when the Dirichlet and Neumann data are known on the same part of the boundary, assuming that the inaccessible part of the boundary is a part of a hyperplane.

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References

  1. Amelinckx S. et al.: Electron Microscopy: Principles and Fundamentals. Weinheim, Wiley–VCH (1997)

    Book  Google Scholar 

  2. Ammari H., Uhlmann G.: Reconstruction of the potential from partial Cauchy data for the Schrödinger equation. Indiana Univ. Math. J. 53(1), 169–183 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arridge S., Lionheart W.: Nonuniqueness in diffusion-based optical tomography. Optics Lett. 23, 882–884 (1998)

    Article  ADS  Google Scholar 

  4. Arridge S.: Optical tomography in medical imaging. Inverse Problems 15, R41 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Astala K., Päivärinta L.: Calderón’s inverse conductivity problem in the plane. Ann. Math. 163, 265–299 (2006)

    Article  MATH  Google Scholar 

  6. Astala K., Lassas M., Päiväirinta L.: Calderón’s inverse problem for anisotropic conductivity in the plane. Comm. Part. Diff. Eqs. 30, 207–224 (2005)

    Article  MATH  Google Scholar 

  7. Ben Joud H.: A stability estimate for an inverse problem for the Schrödinger equation in a magnetic field from partial boundary measurements. Inverse Problems 25(4), 045012 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  8. Brown R., Salo M.: Identifiability at the boundary for first-order terms. Appl. Anal. 85(6–7), 735–749 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bukhgeim A.: Recovering the potential from Cauchy data in two dimensions. J. Inverse Ill-Posed Probl. 16, 19–34 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bukhgeim A., Uhlmann G.: Recovering a potential from partial Cauchy data. Comm. Part. Diff. Eqs. 27(3–4), 653–668 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Calderón, A.: On an inverse boundary value problem. In: Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), Rio de Janeiro: Soc. Brasil. Mat., 1980, pp. 65–73

  12. Case M., Zweifel P.: Linear Transport Theory. Addison-Wesley, New York (1967)

    MATH  Google Scholar 

  13. Chen C.J.: Introduction to scanning tunneling microscopy. Oxford Series in Optical & Imaging Sciences. Oxford Univ. press, Oxford (1993)

    Google Scholar 

  14. Choulli, M.: Une introduction aux problèmes inverses elliptiques et paraboliques. Volume 65 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Berlin:Springer-Verlag, 2009

  15. Colton, D., Kress, R.: Integral equation methods in scattering theory. A Wiley-Interscience Publication. New York: John Wiley and Sons, Inc., 1983

  16. Cristofol, M., Gaitan, P., Iftimie, V.: Inverse problems for the Schrödinger operator in a layer. Rev. Roumaine Math. Pures Appl. 50(2), 153–180 (2005)

    MathSciNet  MATH  Google Scholar 

  17. DosSantos Ferreira D., Kenig C., Sjöstrand J., Uhlmann G.: Determining a magnetic Schrödinger operator from partial Cauchy data. Commun. Math. Phys. 271(2), 467–488 (2007)

    Article  ADS  Google Scholar 

  18. Eskin G., Ralston J.: Inverse scattering problem for the Schrödinger equation with magnetic potential at a fixed energy. Commun. Math. Phys. 173(1), 199–224 (1995)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. Fanelli D., Öktem O.: Electron tomography: a short verview with an emphasis on the absorption potential model for the forward problem. Inverse Problems 24, 013001 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  20. Greenleaf A., Lassas M., Uhlmann G.: The Calderón problem for conormal potentials, I: Global uniqueness and reconstruction. Comm. Pure Appl. Math. 56, 328–352 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Greenleaf A., Lassas M., Uhlmann G.: On nonuniqueness for Calderón’s inverse problem. Math. Res. Lett. 10(5–6), 685–693 (2003)

    MathSciNet  MATH  Google Scholar 

  22. Greenleaf A., Kurylev Y., Lassas M., Uhlmann G.: Full-wave invisibility of active devices at all frequencies. Commun. Math. Phys. 275, 749–789 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. Greenleaf A., Kurylev Y., Lassas M., Uhlmann G.: Invisibility and Inverse Problems. Bull. Amer. Math. Soc. 46, 55–97 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Greenleaf A., Kurylev Y., Lassas M., Uhlmann G.: Cloaking Devices, Electromagnetic Wormholes and Transformation Optics. SIAM Review 51, 3–33 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. Greenleaf A., Kurylev Y., Lassas M., Uhlmann G.: Approximate Quantum and Acoustic Cloaking. J. Spectral Th. 1, 27–80 (2011)

    Article  MathSciNet  Google Scholar 

  26. Grubb, G.: Distributions and operators. Volume 252 of Graduate Texts in Mathematics. New York: Springer, 2009

  27. Hörmander L.: Lower bounds at infinity for solutions of differential equations with constant coefficients. Israel J. Math. 16, 103–116 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hörmander, L.: The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Classics in Mathematics. Berlin: Springer-Verlag, 2003

  29. Ikehata M.: Inverse conductivity problem in the infinite slab. Inverse Problems 17(3), 437–454 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. Isakov, V.: Inverse problems for partial differential equations. Second edition. Applied Mathematical Sciences, 127. New York: Springer, 2006

  31. Isakov V.: On uniqueness in the inverse conductivity problem with local data. Inverse Probl. Imaging 1(1), 95–105 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kachalov, A., Kurylev, Y., Lassas, M.: Inverse Boundary Spectral Problems. Chapman and Hall/CRC Monogr. and Surv. in Pure and Appl. Math. 123, Boca Raton, FL: Chapman and Hall/CRC, 2001

  33. Keijzer M., Star W., Storchi P.: Optical diffusion in layered media. Appl. Opt. 27, 1820–1824 (1988)

    Article  ADS  Google Scholar 

  34. Kenig C., Sjöstrand J., Uhlmann G.: The Calderón problem with partial data. Ann. of Math. (2) 165(2), 567–591 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  35. Knudsen K., Salo M.: Determining nonsmooth first order terms from partial boundary measurements. Inverse Probl. Imaging 1(2), 349–369 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Krupchyk, K., Lassas, M., Uhlmann, G.: Inverse boundary value problems for the perturbed polyharmonic operator. see http://arxiv.org/abs/1102.5542v1 [math.AP], 2011

  37. Lassas M., Uhlmann G.: Determining Riemannian manifold from boundary measurements. Ann. Sci. École Norm. Sup. 34, 771–787 (2001)

    MathSciNet  MATH  Google Scholar 

  38. Lassas M., Taylor M., Uhlmann G.: The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary. Comm. Geom. Anal. 11, 207–222 (2003)

    MathSciNet  MATH  Google Scholar 

  39. Lax, P., Phillips, R.: Scattering theory for the acoustic equation in an even number of space dimensions. Indiana Univ. Math. J. 22, 101–134 (1972/73)

    Google Scholar 

  40. Lee J., Uhlmann G.: Determining anisotropic real-analytic conductivities by boundary measurements. Comm. Pure Appl. Math. 42, 1097–1112 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  41. Li X., Uhlmann G.: Inverse problems with partial data in a slab. Inverse Probl. Imaging 4(3), 449–462 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  42. Morgenröther K., Werner P.: Resonances and standing waves. Math. Methods Appl. Sci. 9(1), 105–126 (1987)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  43. Nachman A.: Reconstructions from boundary measurements. Ann. of Math. 128(2), 531–576 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  44. Nachman A.: Global uniqueness for a two-dimensional inverse boundary value problem. Ann. of Math. 143, 71–96 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  45. Nakamura G., Sun Z., Uhlmann G.: Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field. Math. Ann. 303(3), 377–388 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  46. O’Dell S.: Inverse scattering for the Laplace-Beltrami operator with complex electromagnetic potentials and embedded obstacles. Inverse Problems 22(5), 1579–1603 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  47. Päivärinta L., Panchenko A., Uhlmann G.: Complex geometrical optics for Lipschitz conductivities. Rev. Mat. Iberoam. 19, 57–72 (2003)

    Article  MATH  Google Scholar 

  48. Quinto E.T., Öktem O.: Local Tomography in Electron Microscopy. SIAM J. Appl. Math. 68, 1282–1303 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  49. Reimer, L., Kohl, H.: Transmission electron microscopy: Physics of image formation. Springer Series in Optical Sciences, Berlin-Heidelberg-New York: Springer, 2008

  50. Salo, M.: Inverse problems for nonsmooth first order perturbations of the Laplacian. Ann. Acad. Sci. Fenn. Math. Diss. 139 (2004) available et http://www.rni.helsinki.fi/~msa/pub/thesis.pdf, 2004

  51. Salo M.: Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field. Comm. Part. Diff. Eqs. 31(10–12), 1639–1666 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  52. Salo M., Wang J.-N.: Complex spherical waves and inverse problems in unbounded domains. Inverse Problems 22(6), 2299–2309 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  53. Sun Z.: An inverse boundary value problem for Schrödinger operators with vector potentials. Trans. Amer. Math. Soc. 338(2), 953–969 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  54. Sylvester J., Uhlmann G.: A global uniqueness theorem for an inverse boundary value problem. Ann. of Math. 125, 153–169 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  55. Tolmasky C.F.: Exponentially growing solutions for nonsmooth first–order perturbations of the Laplacian. SIAM J. Math. Anal. 29(1), 116–133 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Helsinki, P. O. Box 68, 00014, Helsinki, Finland

    Katsiaryna Krupchyk & Matti Lassas

  2. Department of Mathematics, University of Washington, Seattle, WA, 98195-4350, USA

    Gunther Uhlmann

  3. Department of Mathematics, University of California, 340 Rowland Hall, Irvine, CA, 92697-3875, USA

    Gunther Uhlmann

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  1. Katsiaryna Krupchyk
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  2. Matti Lassas
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  3. Gunther Uhlmann
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Correspondence to Gunther Uhlmann.

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Communicated by P. Constantin

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Krupchyk, K., Lassas, M. & Uhlmann, G. Inverse Problems With Partial Data for a Magnetic Schrödinger Operator in an Infinite Slab and on a Bounded Domain. Commun. Math. Phys. 312, 87–126 (2012). https://doi.org/10.1007/s00220-012-1431-1

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