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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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##### References:
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