Inverse spectral problem for Sturm–Liouville operator with prescribed partial trace

Abstract

This work is aimed at studying optimization inverse spectral problems with a so-called incomplete spectral data. As incomplete spectral data, the partial traces of the Sturm–Liouville operator serve. We study the following formulation of the inverse spectral problem with incomplete data (optimization problem): find a potential $\hat{V}$ closest to a given function $V_0$ such that a partial trace of the Sturm–Liouville operator with the potential $\hat{V}$ has a prescribed value. As a main result, we prove the existence and uniqueness theorem for solutions of this optimization inverse spectral problem. A new type of relationship between linear spectral problems and systems of nonlinear differential equations is established. This allows us to find a solution to the inverse optimal spectral problem by solving a boundary value problem for a system of nonlinear differential equations and to obtain a solvability of the system of nonlinear differential equations. To prove the uniqueness of solutions, we use the convexity property of the partial trace of the Sturm-Liouville operator with the potential $\hat{V}$; the trace is treated as a functional of the potential $\hat{V}$. We obtain a new generalization of the Lidskii-Wielandt inequality to arbitrary self-adjoint semi-bounded operators with a discrete spectrum.