Determination of a differential pencil from interior spectral data

In this paper, inverse spectra problems for a differential pencil are studied. By using the approach similar to those in Hochstadt and Lieberman (1978) [14] and Ramm (2000) [26], we prove that (1) if p ( x ) (or q ( x ) ) is full given on the interval [ 0 , π ] , then a set of values of eigenfunctions at the mid-point of the interval [ 0 , π ] and one spectrum suffice to determine q ( x ) (or p ( x ) ) on the interval [ 0 , π ] and all parameters in the boundary conditions; (2) if p ( x ) (or q ( x ) ) is full given on the interval [ 0 , π ] , then some information on eigenfunctions at some internal point b ∈ ( π 2 , π ) and parts of two spectra suffice to determine q ( x ) (or p ( x ) ) on the interval [ 0 , π ] and all parameters in the boundary conditions.