Inverse spectral problems for non-self-adjoint Sturm–Liouville operators with discontinuous boundary conditions

Abstract

This paper deals with the inverse spectral problem for a nonself-adjoint Sturm-Liouville operator with discontinuous conditions inside the interval. We obtain that if the potential q is known a priori on a subinterval [b, π] with b ∈ (d, π] or b = d, then h, β, γ and q on [0, π] can be uniquely determined by partial spectral data consisting of a sequence of eigenvalues and a subsequence of the corresponding generalized normalizing constants or a subsequence of the pairs of eigenvalues and the corresponding generalized ratios. For the case b ∈ (0, d) , a similar statement holds if β, γ are also known a priori. Moreover, if q satisfies a local smoothness condition, we provide an alternative approach instead of using the high-energy asymptotic expansion of the Weyl m-function to solve the problem of missing eigenvalues and norming constants.