Ambarzumyan-type theorem for the impulsive Sturm-Liouville operator (Q2660867)

The authors consider a Sturm-Liouville operator on \(\left(0,\frac{\pi}{2}\right)\cup\left(\frac{\pi}{2},\pi\right)\) with an integrable potential \(q\) and a simple step function as a weight. A Neumann boundary condition is imposed at 0 and at \(\pi\), while a discontinuity of the eigenfunctions and their first derivatives is specified at \(\frac{\pi}{2}\). Denoting the eigenvalues by \(\lambda_n(q)\), they prove that \(\lambda_n(q)=\lambda_n(0)\) for all \(n\in\mathbb{N}_0\) implies \(q=0\) a.e. As in Ambarzumyan's original theorem the reason why a single spectrum suffices to determine \(q\) uniquely is the fact that \(\lambda_0(0)\) vanishes.