Neumann series of Bessel functions for inverse coefficient problems

Consider the Sturm-Liouville equation - y '' + q(x) rho(2) y with a real-valued potential q is an element of L-1(0, L), rho is an element of C, L > 0. Let u(rho, x) be its solution satisfying certain initial conditions u(rho(k), 0) = a(k), u'(rho(k), 0) = b(k) for a number of rho(k), k = 1, 2,., K, where rho(k), a(k), and b(k) are some complex numbers. Denote l(k) = u'(rho(k), L) + Hu(rho(k), L), where H is an element of R. We propose a method for solving the inverse problem of the approximate recovery of the potential q(x) and number H from the following data {rho(k), a(k), b(k), l(k)}(k=1)(K). In general, the problem is ill-posed; however, it finds numerous practical applications. Such inverse problems as the recovery of the potential from a Weyl function or the inverse two-spectra Sturm-Liouville problem are its special cases. Moreover, the inverse problem of determining the shape of a human vocal tract also reduces to the considered inverse problem. The proposed method is based on special Neumann series of Bessel functions representations for solutions of Sturm-Liouville equations. With their aid the problem is reduced to the classical inverse Sturm-Liouville problem of recovering q(x) from two spectra, which is solved again with the help of the same representations. The overall approach leads to an efficient numerical algorithm for solving the inverse problem. Its numerical efficiency is illustrated by several examples.