Unique Existence and Reconstruction of the Solution of Inverse Spectral Problem for Differential Pencil

In this paper, the half-inverse spectral problem for energy-dependent Sturm–Liouville problems (that is, differential pencils), defined on interval [0,π] with the potential functions p,q being a priori known on the subinterval [0,π/2], is considered. We provide a method for the unique reconstruction of the two potential functions on [π/2,π] and the boundary condition at x=π by using one full spectrum. Consequently, based on the reconstruction method, we also provide a necessary and sufficient condition under which the existence of the quadratic pencil of differential operators is unique.