C. Maeve McCarthy, The inverse eigenvalue problem for a weighted Helmholtz equation, Applicable Analysis, 77 no. 1-2 (2001), pp. 77-96.
Given the $m$ lowest eigenvalues, we seek to recover an approximation to the density function $\rho$ in the weighted Helmholtz e
equation $-\Delta u =\lambda \rho u$ on a rectangle with Dirchlet boundary conditions. The density $\rho$ is assumed to be symmetric with respect to the midlines of the rectangle. Projection of the boundary value problem and the unknown density function onto appropriate vector spaces leads to a matrix inverse problem. Solutions of the matrix inverse problem exist provided that the reciprocals of the prescribed eigenvalues are close to the reciprocals of the simple eigenvalues of the base problem with $\tho = 1.$ The matrix inverse problem is solved by a fixed-point iterative method and a density function $\rho^*$ is constructed which has the same $m$ lowest eigenvalues as the unknown $\rho.$ The algorithm can be modified when multiple base eigenvalues arise, although the success of the modification depends on the symmetry properties of the base eigenfunctions.