Solution of the inverse problem for the vibronic analogue of the complex Fermi resonance based on the plane Jacobi rotations

Based on algebraic methods, an exact solution is found to the inverse problem for a complex vibronic analogue of the Fermi resonance, which consists in determining from the spectral data for the observed conglomerate of lines (energies $E_k$ and transition intensities $I_k$, $k = 1,2, \dots,n$; $n>$ 2) the energies of the dark states, $A_m$, and the matrix elements $B_m$ of their coupling with the bright state. In the first part of the algorithm, using plane Jacobi rotations, an orthogonal similarity transformation matrix $X$ is found, the first row of which is subject to the condition $(X_{1k})^2=I_{k}$ on its elements, since only one unperturbed state is bright. In the second part, the quantities $A_m$ and $B_m$ are obtained from the solution of the eigenvalue problem for the block of dark states of the matrix $X\operatorname{diag}(\{E_k\})X^{-1}$.