Abstract
In this paper we describe a method for constructing approximate solutions of a two-dimensional inverse eigenvalue problem. Here we consider the problem of recovering a functionq(x, y) from the eigenvalues of — Δ +q(x, y) on a rectangle with Dirichlet boundary conditions. The potentialq(x, y) is assumed to be symmetric with respect to the midlines of the rectangle. Our method is a generalization of an algorithm Hald presented for the construction of symmetric potentials in the one-dimensional inverse Sturm-Liouville problem. Using a projection method, the inverse spectral problem is reduced to an inverse eigenvalue problem for a matrix. We show that if the given eigenvalues are small perturbations of simple eigenvalues ofq=0, then the matrix problem has a solution. This solution is used to construct a functionq which has the same lowest eigenvalues as the unknownq, and several numerical examples are given to illustrate the methods.
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References
Barcilon, V.,A two-dimensional inverse eigenvalue problem, Inverse Problems,6, 11–20 (1990).
Borg, G.,Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Acta Math.,78, 1–96 (1946).
Courant, C. and Hilbert, D.,Methods of Mathematical Physics, Vol. 1, Interscience Publ, New York 1953.
El Badia, A.,On the uniqueness of a bi-dimensional inverse spectral problem, C. R. Acad. Sci. Paris Ser. I Math,308, No. 10, 273–276 (1989).
Friedland, S.,The reconstruction of a symmetric matrix from the spectral data, J. Math. Anal. Appl.,71, 412–422 (1979).
Friedland, S., Nocedal, J., and Overton, M.,The formulation and analysis of numerical methods for inverse eigenvalue problems, SIAM J. Numer. Anal.,24, No. 3, 634–667 (1987).
Hald, O.,The inverse Sturm-Liouville problem and the Rayleigh-Ritz method, Math. Comp.,32, No. 143, 687–705 (1978).
Hald, O.,Inverse eigenvalue problems for the mantle, II, Geophys. J. of Royal Astr. Soc.,72, 139–164 (1983).
Kato, T.,Perturbation Theory for Linear Operators, Springer Verlag, New York 1976.
Kurylev, Y.,Multi-dimensional inverse boundary problems by BC-method: groups of transformations and uniqueness results, to appear J. Math. Comp. Model.
McLaughlin, J. R.,Analytical methods for recovering coefficients in differential equations from spectral data, SIAM Review,28, 53–72 (1986).
Nachman, A., Sylvester, J., and Uhlmann, G.,An n-dimensional Borg-Levinson theorem, Comm. Math. Phys.,115, 595–605 (1988).
Seidman, T.,An inverse eigenvalue problem with rotational symmetry, Inverse Problems,4, 1093–1115 (1988).
Strang, G. and Fix, G. J.,An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ 1973.
Toman, S. and Pliva, J.,Multiplicity of solutions of the inverse secular problem, J. Molecular Spect.,21, 362–371 (1966).
Weinberg, H.,Variational Methods for Eigenvalue Approximation, Regional Conf. Ser. in Appl. Math., Vol 15, SIAM (1974).
Willis, C.,An inverse method using toroidal mode data, Inverse Problems,2, 111–130 (1986).
Yen, A.,Numerical solution of the inverse Sturm-Liouville problem, PhD Thesis, U. of California at Berkeley (1978).
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Knobel, R., McLaughlin, J.R. A reconstruction method for a two-dimensional inverse eigenvalue problem. Z. angew. Math. Phys. 45, 794–826 (1994). https://doi.org/10.1007/BF00942754
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DOI: https://doi.org/10.1007/BF00942754