The inverse nodal problem for a differential operator with an eigenvalue in the boundary condition (Q847397)

The paper considers the Sturm-Liouville problem given by the second-order ODE \[ y''(x)+ [\lambda^2+ \mu- q(x)]y(x)= 0,\qquad x\in (0,\pi)\tag{1} \] subject to the boundary conditions \[ y(0)= 0,\quad ay'(\pi)+\lambda y(\pi)= 0,\tag{2} \] where \(q\in L^1([0,\pi])\), \(\mu\in \mathbb{R}\), \(a\in\mathbb{R}^*\), \(\lambda\) is eigenvalue. It is shown that any dense subset of the nodes (zeros of eigenfunctions) determines uniquely both the constant \(a\in\mathbb{R}^*\) and the potential \[ q(x)- \int^\pi_0 q(\xi)\,d\xi \] for \(x\in (0,\pi)\). It is not very clear what is the role of including the constant parameter \(\mu\in\mathbb{R}\) in equation (1).