Let \(m_j,k_j>0\) for all \(j\in\mathbb{N}\). Consider a self-adjoint extension \(J\) of the minimal symmetric operator \(J_0\) naturally associated in \(l^2(\mathbb{N})\) with the Jacobi (tri-diagonal) matrix \[ J_0=\left(\begin{matrix} q_1 & b_1 & 0 & \dots\\ b_1 & q_2 & b_2 & \dots\\ 0 & b_2 & q_3 & \dots\\ \dots & \dots & \dots & \dots \end{matrix} \right),\quad q_j=-\frac{k_j+k_{j+1}}{m_j},\;b_j=\frac{k_{j+1}}{\sqrt{m_jm_{j+1}}}. \] Note that the corresponding difference equation allows a useful mechanical interpretation. Namely, it describes the motion of a string (sometimes called the Krein-Stieltjes string) with the mass distribution \(M(x)=\sum_{x_j<x}m_j\), where \(x_0:=0\) and \(x_j:=x_{j-1}+k_j\), \(j\in\mathbb{N}\).
Fix \(\theta>0\) and \(h\in\mathbb{R}\) and consider the following operator \(\tilde{J}\), which is a rank 2 perturbation of \(J\): \[ \tilde{J}=J+(q_1(\theta^2-1)+\theta^2h)(e_1,\cdot)e_1+b_1(\theta-1)\big((e_1,\cdot)e_2+(e_2,\cdot)e_1\big). \] Under the assumption that the spectrum of \(J\) (and hence \(\tilde{J}\)) is discrete, the authors investigate the problem of reconstruction of the operator \(J\) from the spectra of \(J\) and \(\tilde{J}\). Also, they present necessary and sufficient conditions for two sequences to be the spectra of \(J\) and \(\tilde{J}\).
For Part I, see [R. del Rio and M. Kudryavtsev, Inverse Probl. 28, No. 5, Article ID 055007 (2012; Zbl 1259.47040), arXiv:1106.1691]. For Part II, see [R. del Rio, M. Kudryavtsev and L. O. Silva, “Inverse problems for Jacobi operators. II: Mass perturbations of semi-infinite mass-spring systems”, arXiv:1106.4598].