Optimization spectral problem for the Sturm–Liouville operator in the space of vector functions

An inverse spectral optimization problem is considered: for a given matrix potential $Q_0(x)$ it is required to find the matrix function $\hat{Q}(x)$ closest to it, such that the $k$-th eigenvalue of the Sturm–Liouville matrix operator with potential $\hat{Q}(x)$ matched the given value $\lambda^*$. The main result of the paper is the proof of existence and uniqueness theorems. Explicit formulas for the optimal potential are established through solutions to systems of nonlinear differential equations of the second order, known in mathematical physics as systems of nonlinear Schrödinger equations