Uniform full stability of recovering convolutional perturbation of the Sturm–Liouville operator from the spectrum

Abstract

Recently, there appeared new stability results in the inverse Sturm–Liouville theory that, unlike the usual local stability, involve uniform estimates. Besides more advanced justification of numerical simulations, the uniform stability can reveal the nature of an inverse problem to a significantly greater extent. For example, it allows obtaining global solvability of an inverse problem as a corollary of its solvability for any dense subclass of input data. In this paper, we obtain uniform stability of an inverse problem for one important class of integro-differential operators, which is a natural generalization of the classical Sturm–Liouville operator. For this purpose, we develop a new method, which is different from that used for the classical problem. Moreover, the considered situation arose relevance of the so-called full stability, i.e. the stability with respect to the full set of the input data including all a priori known components of the operator. The obtained results illustrate some essential difference from the inverse Sturm–Liouville problem.