A numerical method for inverse spectral problems

Basing on the Galerkin methods, we develop a new numerical method for solving the inverse spectral problems generated by discrete lower semibounded operators. The restrictions on the perturbing operator are relaxed in comparison with the method based on the theory of regular traces. A Fredholm integral equation of the first kind enables us to recover the values of the perturbing operator at the discretization nodes. We tested the method on spectral problems for the Sturm–Liouville operator, and the results of numerous simulations demonstrate its computational efficiency.
We found simple formulas for the eigenvalues of a discrete lower semibounded operator avoiding the roots of the corresponding secular equations. The calculation of eigenvalues of these operators can start at an arbitrary index independently of the (un)availability of the eigenvalues with smaller indices. For perturbed selfadjoint operators we can calculate eigenvalues with large indices when the Galerkin method becomes difficult to apply.