The Inverse Eigenvalue Problem with Finite Data for Partial Differential Equations - Barnes, David C.; Knobel, Roger (SIAM journal on mathematical analysis, 1995)
This work is concerned with the inverse eigenvalue problem for the partial differential equation $\nabla ^2 u + (λ - q(x,y))u = 0$ . We study the problem of reconstructing the coefficient function $q(x,y)$ (or at least a numerical approximation to it) using only a finite amount of spectral data, say, $λ _n (q)$ for $n = 1,2, … ,N$. One of the essential tasks considered here is that of determining how much information about the unknown function can be contained in such a fixed and finite amount of spectral data. A numerical method, based on a constrained least squares procedure, is devised for extracting such information, and several examples are given. A proof of convergence for the numerical method is provided. We show that the main difficulty with the finite inverse problem is that the eigenvalues are continuous in some very weak topologies. This work is a higher-dimensional version of the problem considered by Barnes [SIAM J. Math Anal., 22 (1991), pp. 732-753] for ordinary differential equations.