On recovering problem for Sturm — Liouville operator with two frozen arguments

Abstract

Inverse spectral problems consist in recovering operators by their spectral characteristics. The problem of recovering the Sturm — Liouville operator with one frozen argument by one spectrum was considered earlier in works by various authors. In this paper, we study the uniqueness of recovering the operator with two frozen arguments and different coefficients $p$, $q$ by the spectra of two boundary value problems. This case is significantly more difficult than the case of one frozen argument since the operator is no longer a one–dimensional perturbation. We prove that the operator with two frozen arguments can not be recovered by two spectra in the general case. For the unique recovery, one has to impose some conditions on the coefficients. We assume that the coefficients $p$ and $q$ are zero on some interval and prove the uniqueness theorem. We also obtain formulas for regularized traces of two spectra. The result is formulated in terms of the convergence of a certain series, which allows us to avoid smoothness conditions for the coefficients.