Determination of the impulsive Sturm–Liouville operator from a set of eigenvalues

In this work, we consider the inverse spectral problem for the impulsive Sturm–Liouville problem on {(0,\pi)} with the Robin boundary conditions and the jump conditions at the point {\frac{\pi}{2}}. We prove that the potential {M(x)} on the whole interval and the parameters in the boundary conditions and jump conditions can be determined from a set of eigenvalues for two cases: (i) the potential {M(x)} is given on {(0,\frac{(1+\alpha)\pi}{4})}; (ii) the potential {M(x)} is given on {(\frac{(1+\alpha)\pi}{4},\pi)}, where {0<\alpha<1}, respectively. It is also shown that the potential and all the parameters can be uniquely recovered by one spectrum and some information on the eigenfunctions at some interior point.