Inverse problems for the impulsive Sturm–Liouville operator with jump conditions

The inverse problem for impulsive Sturm–Liouville operators with discontinuity conditions is considered. We have shown that all parameters used in the boundary conditions as well as q(x) can be uniquely established by a set of values of eigenfunctions at the mid-point x=π/2 and one spectrum. Moreover, we discuss Gesztesy–Simon theorem and show that if the potential function q(x) is prescribed on the interval [π/2(1−α),π] for some α∈(0,1), then parts of a finite number of spectra suffice to determine q(x) on [0,π].