The Inverse Problem for the Impulsive Differential Pencil

In this paper, we investigate the inverse problem for the impulsive differential pencil in the finite interval. Taking Mochizuki–Trooshin’s theorem, it is proved that two potentials and the boundary conditions are uniquely given by one spectra together with a set of values of eigenfunctions in the situation of \(x=1/2\) . Moreover, applying Gesztesy–Simon’s theorem, we demonstrate that if the potentials are assumed on the interval \([(1-\theta)/2,1],\) where \(\theta\in(0,1),\) a finite number of spectrum are enough to give potentials on \([0,1]\) and other boundary condition.