Recovering differential pencils with spectral boundary conditions and spectral jump conditions

Abstract

In this work, we discuss the inverse problem for second order differential pencils with boundary and jump conditions dependent on the spectral parameter. We establish the following uniqueness theorems: (i) the potentials qk(x) and boundary conditions of such a problem can be uniquely established by some information on eigenfunctions at some internal point b∈(π2,π) and parts of two spectra; (ii) if one boundary condition and the potentials qk(x) are prescribed on the interval [π/ 2 (1 − α) , π] for some α∈ (0 , 1) , then parts of spectra S⊆ σ(L) are enough to determine the potentials qk(x) on the whole interval [0 , π] and another boundary condition.