An inverse spectral problem for a nonsymmetric differential operator: reconstruction of eigenvalue problem (Q860675)

The paper deals with the non-selfadjoint boundary value problem \[ B\mathbf v'(x)+P(x)\mathbf v(x)=\lambda\mathbf v(x),\quad 0<x<1, \] \[ \mathbf v'_2(0)\cos\mu-\mathbf v_1(0)\sin\mu=0,\; \mathbf v'_2(1)\cos\nu+\mathbf v_1(1)\sin\nu=0, \] where \(\lambda\) is the spectral parameter, \[ \mathbf v(x)= \left[ \begin{matrix} \mathbf v_{1}(x) \\ \mathbf v_{2}(x) \end{matrix} \right], \; B= \left[ \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \right],\; P(x)= \left[ \begin{matrix} p_{11}(x) & p_{12}(x) \\ p_{21}(x) & p_{22}(x) \end{matrix} \right], \] \(p_{jk}(x)\in C^1[0,1]\) are complex-valued functions, and \(\mu,\nu\) are complex numbers. The author studies the following inverse problem: Let \(p_{21}(x), p_{22}(x)\) and \(\mu\) are known a priori and fixed. Given the spectral data \(S,\) construct \(p_{11}(x), p_{12}(x)\) and \(\nu\). Using the transformation operator method (which is due to Marchenko and Levitan) the author provides a constructive procedure for the solution of this inverse problem along with necessary and sufficient conditions for its solvability.