Communications in Mathematical Physics

, Volume 115, Issue 4, pp 595–605 | Cite as

Ann-dimensional Borg-Levinson theorem

  • Adrian Nachman
  • John Sylvester
  • Gunther Uhlmann


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 ℝ n . 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.


Boundary Condition Neural Network Statistical Physic Complex System Nonlinear Dynamics 
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  1. [A]
    Agmon, S.: Spectral properties of Schrödinger operators and scattering theory. Ann. Sc. Norm. Super Pisa. (4)2, 151–218 (1975)Google Scholar
  2. [A-H]
    Agmon, S., Hormander, L.: Asymptotic properties of solutions of differential equations with simple characteristics. J. Anal. Math.30, 1–38 (1976)Google Scholar
  3. [B]
    Borg, G.: Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte. Acta. Math.78, 1–96 (1946)Google Scholar
  4. [G-L]
    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. [L]
    Levinson, N.: The inverse Sturm-Liouville problem. Mat. Tidsskr. B. 1949 25–30 (1949)Google Scholar
  6. [L-N]
    Lavine, R. B., Nachman, A. I.: Exceptional points in multidimensional inverse problems (in preparation)Google Scholar
  7. [S-U,I]
    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. [S-U,II]
    Sylvester, J., Uhlmann, G.: A global uniqueness theorem for an inverse boundary value problem. Ann. Math.125, 153–169 (1987)Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Adrian Nachman
    • 1
  • John Sylvester
    • 2
    • 3
  • Gunther Uhlmann
    • 4
  1. 1.Mathematics DepartmentUniversity of RochesterRochesterUSA
  2. 2.Courant Institute of Math. Science, Mathematics DepartmentYale UniversityDurhamUSA
  3. 3.Courant Institute of Math. Science, Mathematics DepartmentMathematics Department Duke UniversityDurhamUSA
  4. 4.Department of MathematicsUniversity of WashingtonSeattleUSA

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