A formula for finding a potential from nodal lines
HTML articles powered by AMS MathViewer
- by Joyce R. McLaughlin and Ole H. Hald PDF
- Bull. Amer. Math. Soc. 32 (1995), 241-247 Request permission
Abstract:
In this announcement we consider an eigenvalue problem which arises in the study of rectangular membranes. The mathematical model is an elliptic equation, in potential form, with Dirichlet boundary conditions. We have shown that the potential is uniquely determined, up to an additive constant, by a subset of the nodal lines of the eigenfunctions. A formula is given which, when the additive constant is fixed, yields an approximation to the potential at a dense set of points. An estimate is presented for the error made by the formula.References
-
L. Friedlander, On certain spectral properties of very weak nonselfadjoint perturbations of selfadjoint operators, Trans. Moscow Math. Soc. 1 (1982), 185-218.
- Leonid Friedlander, On the spectrum of the periodic problem for the Schrödinger operator, Comm. Partial Differential Equations 15 (1990), no. 11, 1631–1647. MR 1079606, DOI 10.1080/03605309908820740
- Joel Feldman, Horst Knörrer, and Eugene Trubowitz, The perturbatively stable spectrum of a periodic Schrödinger operator, Invent. Math. 100 (1990), no. 2, 259–300. MR 1047135, DOI 10.1007/BF01231187
- Ole H. Hald and Joyce R. McLaughlin, Solutions of inverse nodal problems, Inverse Problems 5 (1989), no. 3, 307–347. MR 999065, DOI 10.1088/0266-5611/5/3/008 —, Inverse problems using nodal position data—uniqueness results, algorithms and bounds, Special Program on Inverse Problems (Proc Centre Math. Anal., Austral. Nat. Univ.), vol. 17, Austral. Nat. Univ., Canberra, 1988, pp. 32-59.
- Ole H. Hald and Joyce R. McLaughlin, Inverse nodal problems: finding the potential from nodal lines, Mem. Amer. Math. Soc. 119 (1996), no. 572, viii+148. MR 1370425, DOI 10.1090/memo/0572 T. Kato, Perturbation theory for linear operators, Springer, New York, 1984.
- Joyce R. McLaughlin, Inverse spectral theory using nodal points as data—a uniqueness result, J. Differential Equations 73 (1988), no. 2, 354–362. MR 943946, DOI 10.1016/0022-0396(88)90111-8
- Jürgen K. Moser, Lectures on Hamiltonian systems, Mem. Amer. Math. Soc. 81 (1968), 60. MR 0230498
- K. F. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 1–20; corrigendum, 168. MR 72182, DOI 10.1112/S0025579300000644
- K. Uhlenbeck, Eigenfunctions of Laplace operators, Bull. Amer. Math. Soc. 78 (1972), 1073–1076. MR 319226, DOI 10.1090/S0002-9904-1972-13117-3
- K. Uhlenbeck, Generic properties of eigenfunctions, Amer. J. Math. 98 (1976), no. 4, 1059–1078. MR 464332, DOI 10.2307/2374041 A. Weinman-Romer, Stabile Eigenwerte des Laplace-Operators auf Speziellen Tori, Diplomarbeit, ETH, Zürich, 1989.
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 32 (1995), 241-247
- MSC: Primary 35R30; Secondary 35J99, 35P20, 73D50, 73K10
- DOI: https://doi.org/10.1090/S0273-0979-1995-00584-7
- MathSciNet review: 1302784