On a Spectral Inverse Problem in Perturbation Theory

We consider an inverse spectral problem for Sturm-Liouville operators (H) over cap (V) defined on the interval [a, b] by a certain potential V is an element of L-2[a, b] and mixed separated boundary conditions. We show that if the L-1-norm of V is small enough, then there exists V-app such that parallel to V - V-app parallel to(L2) = O(parallel to V parallel to(2)(L1)) and we indicate an algorithm to find V-app. The algorithm determines the Fourier coefficients of V-app with respect to eigenfunctions {psi(k,0)}(k=1)(infinity) of the unperturbed operator (H) over cap (0) via eigenvalues {lambda(k,V)}(k=1)(infinity) of the "perturbed" operator (H) over cap (V), the values of its eigenfunctions {psi(k,V)}(k=1)(infinity) at the endpoints of [a, b], and values of {psi(k,V)}(k=1)(infinity) and their derivatives at the middle of [a, b].