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Determining a Magnetic Schrödinger Operator from Partial Cauchy Data

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Abstract

In this paper we show, in dimension n ≥ 3, that knowledge of the Cauchy data for the Schrödinger equation in the presence of a magnetic potential, measured on possibly very small subsets of the boundary, determines uniquely the magnetic field and the electric potential. We follow the general strategy of [7] using a richer set of solutions to the Dirichlet problem that has been used in previous works on this problem.

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References

  1. Boman J., Quinto T. (1987) Support theorems for real-analytic Radon transforms. Duke Math. J. 55(4): 943–948

    Article  MATH  MathSciNet  Google Scholar 

  2. Boman J.: Helgason’s support theorem for Radon transforms – a new proof and a generalization. In: Mathematical methods in tomography (Oberwolfach, 1990), Lecture Notes in Math. 1497, Berlin- Heidelberg-NewYork: Springer, 1991, pp. 1–5

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

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. Helgason S.: The Radon transform. 2nd ed., Progress in Math., Basel-Boston: Birkhäuser, 1999

  5. Hörmander L. (1993) Remarks on Holmgren’s uniqueness theorem. Ann. Inst. Fourier 43(5): 1223–1251

    MATH  MathSciNet  Google Scholar 

  6. Hörmander L., (1990) The Analysis of Linear Partial Differential Operators Classics in Mathematics. Berlin-Heidelberg-New York, Springer

    Google Scholar 

  7. Kenig C.E., Sjöstrand J., Uhlmann G. The Calderón problem with partial data. To appear in Ann. of Math.

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Novikov R.G., Khenkin G.M. (1987) The \({\overline\partial}\) -equation in the multidimensional inverse scattering problem. Russ. Math. Surv. 42, 109–180

    Article  MATH  MathSciNet  Google Scholar 

  10. Salo M.: Inverse problems for nonsmooth first order perturbations of the Laplacian. Ann. Acad. Scient. Fenn. Math. Dissertations, Vol. 139, 2004

  11. Salo M. Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field. To appear in Comm. in PDE

  12. Sun Z. (1992) An inverse boundary value problem for the Schrödinger operator with vector potentials. Trans. Amer. Math. Soc. 338(2): 953–969

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to David Dos Santos Ferreira.

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Communicated by P. Constantin

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Ferreira, D.D.S., Kenig, C.E., Sjöstrand, J. et al. Determining a Magnetic Schrödinger Operator from Partial Cauchy Data. Commun. Math. Phys. 271, 467–488 (2007). https://doi.org/10.1007/s00220-006-0151-9

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  • DOI: https://doi.org/10.1007/s00220-006-0151-9

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