Inverse Problem for Differential Pencils with Incompletely Spectral Information

In this paper we are concerned with the inverse spectral problems for energy-dependent Sturm-Liouville problems (that is, differential pencils) defined on interval [0, 1] with two potentials known on a subinterval [a1, a2] ⊂ [0, 1]. We prove that the potentials on the entire interval [0, 1] and the boundary condition at x = 1 are uniquely determined in terms of partial knowledge of the spectrum in the situation of a1 = 0 and a2 ≥ 1/2. Furthermore, in the situation of a1 > 0 and 1/2 ∈ [a1, a2] we need additional information on the eigenfunctions at some interior point to obtain the uniqueness of the potentials on [0, 1] and two boundary conditions at x = 0, 1.