Inverse problems for discontinuous Sturm-Liouville operators with mixed spectral data

Abstract

In this paper, we discuss the inverse spectral problem for Sturm–Liouville operators with boundary conditions linearly dependent on the spectral parameter and a finite number of interior discontinuities and show that if q is given a priori on the interval [0,α0] for some α0∈[0,1), then the potential q on the whole interval [0,1] can be uniquely determined either by parts of a finite number of spectra, or by a finite number of subsets of pairs of eigenvalues and the norming constants of the corresponding eigenvalues. We still establish several uniqueness theorems for Sturm–Liouville operators with Robin boundary conditions and interior discontinuous conditions from the above two spectral data.