INVERSE PROBLEM FOR DIFFERENTIAL PENCILS WITH INCOMPLETELY SPECTRAL INFORMATION

Abstract

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 $ [a_{1},a_{2}]\subset [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 $a_{1}=0$ and $a_{2}\geq 1/2$. Furthermore, in the situation of $a_{1}\gt 0$ and $1/2\in [a_{1},a_{2}]$ 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 $x=0,1$.