Uniform full stability of recovering convolutional perturbation of the Sturm-Liouville operator from the spectrum (Q2656236)

The author studies the uniform stability of an inverse problem for an important class of integro-differential operators which is a natural generalization of the classical Sturm-Liouville operator. For this purpose, he develops a new method which is different from that used for the classical problem. This situation has relevance in the so-called full stability problem, that is, the stability with respect to the full set of the input data including all a priori known components of the operator. The inverse problem considered here consists in recovering the square-integrable potential \( q(x) \) from the spectra \( \{ \lambda_{nj} \}_{n\in N} \) of two eigenvalue problems \( L_{j}(q) \), \(j=0,1\), with one common boundary condition such as \[ L_{j}(q): -y'' +q(x)y=\lambda y,\quad 0< x < \pi , \] \[ y(0)=y^{(j)}(\pi)=0. \] The uniform stability can reveal the nature of an inverse problem to a greater extent. In this connection, the author discusses an abstract scheme to obtain the global solvability of an inverse problem as a corollary from its uniform stability and solvability on any dense subset of the input data, provided that the corresponding direct problem is stable at least locally.