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A reconstruction method for a two-dimensional inverse eigenvalue problem. (English) Zbl 0828.35140
Given \(\{\lambda_k : k = 1, \ldots, m\}\), the authors consider the problem of recovering a function \(q\) defined on \(R = ] 0, \pi/a [\times ]0, \pi[\) such that \(q (\pi/a - x,y) = q(x,y) = q(x, \pi - y)\) and such that \(\{\lambda_k : k = 1, \ldots, m\}\) is the set of eigenvalues of \(- \Delta u + qu = \lambda u\) in \(R\), \(u = 0\) on \(\partial R\) and, using the Rayleigh-Ritz approach of computing eigenvalues, the problem is reduced to an inverse eigenvalue problem for a matrix, as the entries of the matrix are linear combinations of Fourier coefficients of \(q\). The problem is solved if \(\lambda_k (k = 1, \ldots, m)\) are sufficiently close to the smallest \(m\) eigenvalues of the problem with \(q = 0\) and such \(m\) eigenvalues are simple or, if multiple, satisfy a convenient hypothesis.
Reviewer: G.Bottaro (Genova)

MSC:
35R30 Inverse problems for PDEs
35P05 General topics in linear spectral theory for PDEs
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[1] Barcilon, V.,A two-dimensional inverse eigenvalue problem, Inverse Problems,6, 11-20 (1990). · Zbl 0712.35105 · doi:10.1088/0266-5611/6/1/004
[2] Borg, G.,Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Acta Math.,78, 1-96 (1946). · Zbl 0063.00523 · doi:10.1007/BF02421600
[3] Courant, C. and Hilbert, D.,Methods of Mathematical Physics, Vol. 1, Interscience Publ, New York 1953. · Zbl 0051.28802
[4] 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). · Zbl 0679.35083
[5] Friedland, S.,The reconstruction of a symmetric matrix from the spectral data, J. Math. Anal. Appl.,71, 412-422 (1979). · Zbl 0421.65027 · doi:10.1016/0022-247X(79)90201-4
[6] 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). · Zbl 0622.65030 · doi:10.1137/0724043
[7] Hald, O.,The inverse Sturm-Liouville problem and the Rayleigh-Ritz method, Math. Comp.,32, No. 143, 687-705 (1978). · Zbl 0432.65050 · doi:10.1090/S0025-5718-1978-0501963-2
[8] Hald, O.,Inverse eigenvalue problems for the mantle, II, Geophys. J. of Royal Astr. Soc.,72, 139-164 (1983). · Zbl 0507.73085
[9] Kato, T.,Perturbation Theory for Linear Operators, Springer Verlag, New York 1976. · Zbl 0342.47009
[10] Kurylev, Y.,Multi-dimensional inverse boundary problems by BC-method: groups of transformations and uniqueness results, to appear J. Math. Comp. Model. · Zbl 0818.35138
[11] McLaughlin, J. R.,Analytical methods for recovering coefficients in differential equations from spectral data, SIAM Review,28, 53-72 (1986). · Zbl 0589.34024 · doi:10.1137/1028003
[12] Nachman, A., Sylvester, J., and Uhlmann, G.,An n-dimensional Borg-Levinson theorem, Comm. Math. Phys.,115, 595-605 (1988). · Zbl 0644.35095 · doi:10.1007/BF01224129
[13] Seidman, T.,An inverse eigenvalue problem with rotational symmetry, Inverse Problems,4, 1093-1115 (1988). · Zbl 0687.35108 · doi:10.1088/0266-5611/4/4/011
[14] Strang, G. and Fix, G. J.,An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ 1973. · Zbl 0356.65096
[15] Toman, S. and Pliva, J.,Multiplicity of solutions of the inverse secular problem, J. Molecular Spect.,21, 362-371 (1966). · doi:10.1016/0022-2852(66)90162-7
[16] Weinberg, H.,Variational Methods for Eigenvalue Approximation, Regional Conf. Ser. in Appl. Math., Vol 15, SIAM (1974).
[17] Willis, C.,An inverse method using toroidal mode data, Inverse Problems,2, 111-130 (1986). · Zbl 0593.73029 · doi:10.1088/0266-5611/2/1/009
[18] Yen, A.,Numerical solution of the inverse Sturm-Liouville problem, PhD Thesis, U. of California at Berkeley (1978).
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