Article
Ufa Mathematical Journal
Volume 12, Number 4, pp. 19-29
Inverse spectral problem for Sturm-Liouville operator with prescribed partial trace
Il'yasov Y.Sh., Valeev N.F.
DOI:10.13108/2020-12-4-19
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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.