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Inverse vector-valued Sturm-Liouville problem. I. Uniqueness theorem

This paper starts a series devoted to the vector-valued Sturm-Liouville problem $-ψ''+V(x)ψ=λψ$, $ψ\in L^2([0,1];\mathbb{C}^N)$, with separated boundary conditions. The overall goal of the series is to give a complete characterization of classes of spectral data corresponding to potentials $V=V^*\in L^p([0,1];\mathbb{C}^{N\times N})$ for a fixed $1\le p <+\infty$ and separated boundary conditions having the most general form. In the first paper we briefly describe our approach to this inverse problem and prove some preliminary results including the relevant uniqueness theorem.