Given a finite set of eigenvalues of a regular Sturm-Liouville problem for
the equation -y{\prime}{\prime}+q(x)y={\lambda}y, the potential q(x) of which
is unknown. We show the possibility to compute more eigenvalues without any
additional information on the potential q(x). Moreover, considering the
Sturm-Liouville problem with the boundary conditions y{\prime}(0)-hy(0)=0 and
y{\prime}({\pi})+Hy({\pi})=0, where h, H are some constants, we complete its
spectrum without additional information neither on the potential q(x) nor on
the constants h and H. The eigenvalues are computed with a uniform absolute
accuracy. Based on this result we propose a new method for numerical solution
of the inverse Sturm-Liouville problem of recovering the potential from two
spectra. The method includes the completion of the spectra in the first step
and reduction to a system of linear algebraic equations in the second. The
potential q(x) is recovered from the first component of the solution vector.
The approach is based on special Neumann series of Bessel functions
representations for solutions of Sturm-Liouville equations possessing
remarkable properties and leads to an efficient numerical algorithm for solving
inverse Sturm-Liouville problems.
