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Ann-dimensional Borg-Levinson theorem

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Abstract

We show that the potentialq is uniquely determined by the spectrum, and boundary values of the normal derivatives of the eigenfunctions of the Schrödinger operator −Δ+q with Dirichlet boundary conditions on a bounded domain Ω in ℝn. This and related results can be viewed as a direct generalization of the theorem in the title, which states that the spectrum and the norming constants determine the potential in the one dimensional case.

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References

  1. Agmon, S.: Spectral properties of Schrödinger operators and scattering theory. Ann. Sc. Norm. Super Pisa. (4)2, 151–218 (1975)

    Google Scholar 

  2. Agmon, S., Hormander, L.: Asymptotic properties of solutions of differential equations with simple characteristics. J. Anal. Math.30, 1–38 (1976)

    Google Scholar 

  3. Borg, G.: Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte. Acta. Math.78, 1–96 (1946)

    Google Scholar 

  4. Gelfand, I. M. Levitan, B. M.: On the determination of a differential equation from its spectral function. Izv. Akad Nauk. SSSR, Ser. Mat.15, 309–360 (1961)

    Google Scholar 

  5. Levinson, N.: The inverse Sturm-Liouville problem. Mat. Tidsskr. B. 1949 25–30 (1949)

    Google Scholar 

  6. Lavine, R. B., Nachman, A. I.: Exceptional points in multidimensional inverse problems (in preparation)

  7. Sylvester, J., Uhlmann, G.: A uniqueness theorem for an inverse boundary value problem in electrical prospection. Commun. Pure. Appl. Math.39, 91–112 (1986)

    Google Scholar 

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

    Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, University of Rochester, 14627, Rochester, NY, USA

    Adrian Nachman

  2. Courant Institute of Math. Science, Mathematics Department, Yale University, NC27706, Durham, USA

    John Sylvester

  3. Courant Institute of Math. Science, Mathematics Department, Mathematics Department Duke University, NC27706, Durham, USA

    John Sylvester

  4. Department of Mathematics, University of Washington, 98195, Seattle, WA, USA

    Gunther Uhlmann

Authors
  1. Adrian Nachman
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  2. John Sylvester
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  3. Gunther Uhlmann
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Additional information

Communicated by C. H. Taubes

Supported by NSF grant DMS-8602033

Supported by NSF grant DMS-8600797

Supported by NSF grant DMS-8601118 and an Alfred P. Sloan Research Fellowship

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Nachman, A., Sylvester, J. & Uhlmann, G. Ann-dimensional Borg-Levinson theorem. Commun.Math. Phys. 115, 595–605 (1988). https://doi.org/10.1007/BF01224129

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  • DOI: https://doi.org/10.1007/BF01224129

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