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 & 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.  doi: 10.4007/annals.2006.163.265.  Google Scholar

[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.   Google Scholar

[3]

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

[4]

R. Beals and R. R. Coifman, Multidimensional inverse scatterings and nonlinear partial differential equations,, in, 43 (1985), 45.   Google Scholar

[5]

R. Beals and R. R. Coifman, The spectral problem for the Davey-Stewartson and Ishimori hierarchies,, in, (1988), 15.   Google Scholar

[6]

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

[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.  doi: 10.1080/03605309708821292.  Google Scholar

[8]

A. P. Calderón, On an inverse boundary value problem,, in, (1980).   Google Scholar

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

[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.  doi: 10.1137/S0036141003422497.  Google Scholar

[11]

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

[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.  doi: 10.1007/s00041-004-8011-5.  Google Scholar

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[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.  doi: 10.1002/cpa.3160390106.  Google Scholar

[25]

C. F. 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

[26]

G. Uhlmann, Developments in inverse problems since Calderón's foundational paper,, Chapter 19 in, (1999), 295.   Google Scholar

[27]

G. Uhlmann, Electrical impedance tomography and Calderón's problem,, Inverse Problems., ().   Google Scholar

[28]

I. N. Vekua, Generalized analytic functions,, 2nd ed., (1962).   Google Scholar

show all references

References:
[1]

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

[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.   Google Scholar

[3]

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

[4]

R. Beals and R. R. Coifman, Multidimensional inverse scatterings and nonlinear partial differential equations,, in, 43 (1985), 45.   Google Scholar

[5]

R. Beals and R. R. Coifman, The spectral problem for the Davey-Stewartson and Ishimori hierarchies,, in, (1988), 15.   Google Scholar

[6]

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

[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.  doi: 10.1080/03605309708821292.  Google Scholar

[8]

A. P. Calderón, On an inverse boundary value problem,, in, (1980).   Google Scholar

[9]

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

[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.  doi: 10.1137/S0036141003422497.  Google Scholar

[11]

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

[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.  doi: 10.1007/s00041-004-8011-5.  Google Scholar

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[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.  doi: 10.1002/cpa.3160390106.  Google Scholar

[25]

C. F. 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

[26]

G. Uhlmann, Developments in inverse problems since Calderón's foundational paper,, Chapter 19 in, (1999), 295.   Google Scholar

[27]

G. Uhlmann, Electrical impedance tomography and Calderón's problem,, Inverse Problems., ().   Google Scholar

[28]

I. N. Vekua, Generalized analytic functions,, 2nd ed., (1962).   Google Scholar

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