Inverse Problems for Sturm-Liouville Operators with Potentials in Sobolev Spaces: Uniform Stability

Two inverse problems for the Sturm-Liouville operator Ly = -y '' + q(x)y on the interval [0, p] are studied. For theta >= 0, there is a mapping F: W-2(theta). l(B)(theta), F(sigma) = {s(k)}(1)(infinity), related to the first of these problems, where W-2(B) = W-2(theta)[0, pi] is the Sobolev space, sigma = integral q is a primitive of the potential q, and l(B)(theta) is a specially constructed finite-dimensional extension of the weighted space l(2)(theta), where we place the regularized spectral data s = {sk}(1)(infinity) in the problem of reconstruction from two spectra. The main result is uniform lower and upper bounds for parallel to sigma - sigma(1)parallel to(theta) via the l(B)(theta)-norm parallel to s - s(1)parallel to. of the difference of regularized spectral data. A similar result is obtained for the second inverse problem, that is, the problem of reconstructing the potential from the spectral function of the operator L generated by the Dirichlet boundary conditions. The result is new even for the classical case q is an element of L-2, which corresponds to theta = 1.