Inverse nodal problem for diffusion operator on a star graph with nonhomogeneous edges (Q6554747)

The paper deals with the Sturm-Liouville differential pencil on the star-shaped graph: \N\[\N -y_j''(x) + [2\lambda p_j(x) + q_j(x)]y_j(x) = \lambda^2 y_j(x), \quad x \in \left(0, \tfrac{\ell_j}{2}\right) \cap \left(\tfrac{\ell_j}{2}, \ell_j\right), \quad j = \overline{1,\nu}, \N\] \Nsubject to the standard boundary conditions at the internal vertex and the Robin boundary conditions. Additionally, the discontinuity conditions are imposed in the middle point of each edge: \N\begin{align*}\Ny_j\left(\tfrac{\ell_j}{2} + 0\right) &= \alpha_j y_j\left(\tfrac{\ell_j}{2}-0\right), \\\Ny_j'\left(\tfrac{\ell_j}{2} + 0\right) &= \alpha_j^{-1} y_j'\left(\tfrac{\ell_j}{2}-0\right) + \beta_j y_j\left(\tfrac{\ell_j}{2}-0\right).\N\end{align*}\NHere, \(\lambda\) is the spectral parameter, \(p_j \in W_2^1[0,\ell_j]\) and \(q_j \in L_2[0,\ell_j]\) are real-valued functions, \(\alpha_j > 0\) and \(\beta_j\) are reals.\N\NThe author studies the inverse nodal problem, which consists in the recovery of the differential pencil parameters \(p_j\), \(q_j\) and the coefficients of the boundary conditions from the nodal points. The standard method, which is based on the asymptotics of the eigenvalues and the nodal points, is applied.