Inverse problem for Dirac system with singularities in interior points (Q281021)

The eigenvalue problem for the 1-dimensional Dirac system \[ JY'+(Q_w+Q)Y=\lambda Y\quad\text{on}\quad [0,\pi] \] with the boundary conditions \[ [\cos\alpha,\sin\alpha]Y(0)=0=[\cos\beta,\sin\beta]Y'(0) \] is considered. Here, \[ J=\begin{bmatrix} 0&1\\ -1 &0\end{bmatrix}, Q=\begin{bmatrix} q_1&q_2\\ q_2 &q_1\end{bmatrix} \text{ and } Q_w=\frac{\mu_k}{x-\gamma_k}\begin{bmatrix} \sin 2\eta_k&\cos 2\eta_k\\ \cos 2\eta_k &\sin 2\eta_k\end{bmatrix} \] for \(x\in W_k, \eta_k\in \mathbb{R}\). In the above, \(0<\gamma_1<\gamma_2<\dots<\gamma_N<\pi\) and \(W_{k}, k=1,\dots,N\) is a partition of \([0,\pi]\) into \(N\) disjoint intervals with \(\gamma_k\) in the interior of \(W_k\). In addition, the \(\mu_k\) are complex constants and the functions \(q_1\) and \(q_2\) are absolutely continuous complex valued functions with \[ q_j(x)\prod_{k=1}^N |x-\gamma_k|^{-2\Re \mu_k}\in L^1(0,\pi). \] The eigenvalue prolem is discussed as well as uniqueness results for various inverse spectral problems. A procedure is given for the reconstruction of the potential as well as necessary and sufficient conditions for the solvability of one of the inverse problems are given.