February  2011, 5(1): 59-73. doi: 10.3934/ipi.2011.5.59

Global uniqueness for an inverse problem for the magnetic Schrödinger operator

1. 

Department of Mathematics, University of Washington, Seattle, Washington 98105, United States

Received  December 2009 Revised  November 2010 Published  February 2011

In this paper, we prove the global uniqueness of determining both the magnetic field and the electrical potential by boundary measurements in two-dimensional case. In other words, we prove the uniqueness of this inverse problem without any smallness assumption.
Citation: Ru-Yu Lai. Global uniqueness for an inverse problem for the magnetic Schrödinger operator. Inverse Problems and Imaging, 2011, 5 (1) : 59-73. doi: 10.3934/ipi.2011.5.59
References:
[1]

K. Astala and L. Pävärinta, Calderón's inverse conductivity problem in the plane, Annals of Math., 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265.

[2]

R. M. Brown, Recovering the conductivity at the boundary from the Dirichlet to Neumann map: A pointwise result, J. Inv. Ill-Posed Problems, 9 (2001), 567-574.

[3]

A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inv. Ill-Posed Problems, 16 (2008), 19-33. doi: 10.1515/jiip.2008.002.

[4]

R. Beals and R. R. Coifman, Multidimensional inverse scatterings and nonlinear partial differential equations, in "Pseudodifferential Operators and Applications" (F. Treves, editor), Proceedings of Symposia in Pure Mathematics, 43, Amer. Math. Soc., (1985), 45-70.

[5]

R. Beals and R. R. Coifman, The spectral problem for the Davey-Stewartson and Ishimori hierarchies, in "Nonlinear Evolution Equations: Integrability and Spectral Methods," Manchester University Press, (1988), 15-23.

[6]

R. M. Brown and M. Salo, Identifiability at the boundary for first-order terms, Applicable Analysis, 85 (2006), 735-749. doi: 10.1080/00036810600603377.

[7]

R. M. Brown and G. Uhlmann, Uniquenes in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Comm. in Partial Differential Equations, 22 (1997), 1009-1027. doi: 10.1080/03605309708821292.

[8]

A. P. Calderón, On an inverse boundary value problem, in "Seminar on Numerical Analysis and its Applications to Continuum Physics," Rí o de Janeiro, (1980).

[9]

J. Cheng, G. Nakamura and E. Somersalo, Uniquenss of identifying the convection term, Comm. Korean Math. Soc., 16 (2001), 405-413.

[10]

J. Cheng and M. Yamamoto, Determination of two convection coefficients from Dirichlet to Neumann map in the two-dimensional case, SIAM J. Math. Anal., 35 (2004), 1371-1393. doi: 10.1137/S0036141003422497.

[11]

G. Eskin and J. Ralston, Inverse scattering problem for the Schr$\ddot{O}$dinger equation with magnetic potential at a fixed energy, Comm. Math. Phys., 173 (1995), 199-224. doi: 10.1007/BF02100187.

[12]

H. Kang and G. Uhlmann, Inverse problems for the Pauli Hamiltonian in two dimensions, The Journal of Fourier Analysis and Applications, 10 (2004), 201-215. doi: 10.1007/s00041-004-8011-5.

[13]

R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Comm. Pure Appl. Math., 37 (1984), 289-298. doi: 10.1002/cpa.3160370302.

[14]

A. I. Nachman, Reconstructions from boundary measurements, Annals of Math., 128 (1988), 531-576. doi: 10.2307/1971435.

[15]

A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals of Math., 143 (1996), 71-96. doi: 10.2307/2118653.

[16]

G. Nakamura, Z. Sun and G. Uhlmann, Global identifiability for an inverse problem for the Schr$\ddot{O}$dinger equation in a magnetic field, Math. Ann., 303 (1995), 377-388. doi: 10.1007/BF01460996.

[17]

L. Nirenberg and H. F. Walker, The null spaces of elliptic partial differential operators in $\mathbb{R}^{N}$, Journal of Math. Analysis and Applications, 42 (1973), 271-301. doi: 10.1016/0022-247X(73)90138-8.

[18]

M. Salo, Semiclassical pseudodifferential calculus and the reconstuction of a magnetic field, Comm. in PDE, 31 (2006), 1639-1666. doi: 10.1080/03605300500530420.

[19]

M. Salo, Inverse Problems for Nonsmooth First Order Perturbations of the Laplacian, Ann. Acad. Sci. Fenn. Math. Dissertationes, 139 (2004), 67.

[20]

Z. Sun, An inverse boundary value problem for Schrödinger operators with vector potentials, Trans. of AMS, 338 (1993), 953-971. doi: 10.2307/2154438.

[21]

Z. Sun, An inverse boundary value problem for Schrödinger operators with vector potentials in two dimensions, Comm. in PDE, 18 (1993), 83-124. doi: 10.1080/03605309308820922.

[22]

J. Sylveser, An anisotropic inverse boundary value problem, Comm. on Pure and Applied Math., 43 (1990), 201-232. doi: 10.1002/cpa.3160430203.

[23]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Annals of Math., 125 (1987), 153-169. doi: 10.2307/1971291.

[24]

J. Sylvester and G. Uhlmann, A uniqueness theorem for an inverse boundary value problem in electrical prospection, Comm. Pure and Applied Math., 39 (1986), 91-112. doi: 10.1002/cpa.3160390106.

[25]

C. F. Tolmasky, Exponentially growing solutions for nonsmooth first-order perturbations of the laplacian, SIAM J. Math. Anal., 29 (1998), 116-133. doi: 10.1137/S0036141096301038.

[26]

G. Uhlmann, Developments in inverse problems since Calderón's foundational paper, Chapter 19 in "Harmonic Analysis and Partial Differential Equations" (edited by M. Christ, C. Kenig and C. Sadosky), University of Chicago Press, (1999), 295-345,

[27]

G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems.

[28]

I. N. Vekua, Generalized analytic functions, 2nd ed., Pergamon Press, London, 1962.

show all references

References:
[1]

K. Astala and L. Pävärinta, Calderón's inverse conductivity problem in the plane, Annals of Math., 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265.

[2]

R. M. Brown, Recovering the conductivity at the boundary from the Dirichlet to Neumann map: A pointwise result, J. Inv. Ill-Posed Problems, 9 (2001), 567-574.

[3]

A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inv. Ill-Posed Problems, 16 (2008), 19-33. doi: 10.1515/jiip.2008.002.

[4]

R. Beals and R. R. Coifman, Multidimensional inverse scatterings and nonlinear partial differential equations, in "Pseudodifferential Operators and Applications" (F. Treves, editor), Proceedings of Symposia in Pure Mathematics, 43, Amer. Math. Soc., (1985), 45-70.

[5]

R. Beals and R. R. Coifman, The spectral problem for the Davey-Stewartson and Ishimori hierarchies, in "Nonlinear Evolution Equations: Integrability and Spectral Methods," Manchester University Press, (1988), 15-23.

[6]

R. M. Brown and M. Salo, Identifiability at the boundary for first-order terms, Applicable Analysis, 85 (2006), 735-749. doi: 10.1080/00036810600603377.

[7]

R. M. Brown and G. Uhlmann, Uniquenes in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Comm. in Partial Differential Equations, 22 (1997), 1009-1027. doi: 10.1080/03605309708821292.

[8]

A. P. Calderón, On an inverse boundary value problem, in "Seminar on Numerical Analysis and its Applications to Continuum Physics," Rí o de Janeiro, (1980).

[9]

J. Cheng, G. Nakamura and E. Somersalo, Uniquenss of identifying the convection term, Comm. Korean Math. Soc., 16 (2001), 405-413.

[10]

J. Cheng and M. Yamamoto, Determination of two convection coefficients from Dirichlet to Neumann map in the two-dimensional case, SIAM J. Math. Anal., 35 (2004), 1371-1393. doi: 10.1137/S0036141003422497.

[11]

G. Eskin and J. Ralston, Inverse scattering problem for the Schr$\ddot{O}$dinger equation with magnetic potential at a fixed energy, Comm. Math. Phys., 173 (1995), 199-224. doi: 10.1007/BF02100187.

[12]

H. Kang and G. Uhlmann, Inverse problems for the Pauli Hamiltonian in two dimensions, The Journal of Fourier Analysis and Applications, 10 (2004), 201-215. doi: 10.1007/s00041-004-8011-5.

[13]

R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Comm. Pure Appl. Math., 37 (1984), 289-298. doi: 10.1002/cpa.3160370302.

[14]

A. I. Nachman, Reconstructions from boundary measurements, Annals of Math., 128 (1988), 531-576. doi: 10.2307/1971435.

[15]

A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals of Math., 143 (1996), 71-96. doi: 10.2307/2118653.

[16]

G. Nakamura, Z. Sun and G. Uhlmann, Global identifiability for an inverse problem for the Schr$\ddot{O}$dinger equation in a magnetic field, Math. Ann., 303 (1995), 377-388. doi: 10.1007/BF01460996.

[17]

L. Nirenberg and H. F. Walker, The null spaces of elliptic partial differential operators in $\mathbb{R}^{N}$, Journal of Math. Analysis and Applications, 42 (1973), 271-301. doi: 10.1016/0022-247X(73)90138-8.

[18]

M. Salo, Semiclassical pseudodifferential calculus and the reconstuction of a magnetic field, Comm. in PDE, 31 (2006), 1639-1666. doi: 10.1080/03605300500530420.

[19]

M. Salo, Inverse Problems for Nonsmooth First Order Perturbations of the Laplacian, Ann. Acad. Sci. Fenn. Math. Dissertationes, 139 (2004), 67.

[20]

Z. Sun, An inverse boundary value problem for Schrödinger operators with vector potentials, Trans. of AMS, 338 (1993), 953-971. doi: 10.2307/2154438.

[21]

Z. Sun, An inverse boundary value problem for Schrödinger operators with vector potentials in two dimensions, Comm. in PDE, 18 (1993), 83-124. doi: 10.1080/03605309308820922.

[22]

J. Sylveser, An anisotropic inverse boundary value problem, Comm. on Pure and Applied Math., 43 (1990), 201-232. doi: 10.1002/cpa.3160430203.

[23]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Annals of Math., 125 (1987), 153-169. doi: 10.2307/1971291.

[24]

J. Sylvester and G. Uhlmann, A uniqueness theorem for an inverse boundary value problem in electrical prospection, Comm. Pure and Applied Math., 39 (1986), 91-112. doi: 10.1002/cpa.3160390106.

[25]

C. F. Tolmasky, Exponentially growing solutions for nonsmooth first-order perturbations of the laplacian, SIAM J. Math. Anal., 29 (1998), 116-133. doi: 10.1137/S0036141096301038.

[26]

G. Uhlmann, Developments in inverse problems since Calderón's foundational paper, Chapter 19 in "Harmonic Analysis and Partial Differential Equations" (edited by M. Christ, C. Kenig and C. Sadosky), University of Chicago Press, (1999), 295-345,

[27]

G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems.

[28]

I. N. Vekua, Generalized analytic functions, 2nd ed., Pergamon Press, London, 1962.

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