Inverse problems for the Sturm–Liouville operator with discontinuity

Abstract

In the previous works of the author (Applied Mathematics Letters. 2009;22:1315–1319, and Integral Equations and Operator Theory. 2009;65:593–604), provided that discontinuity parameters and boundary condition parameters are known a priori the inverse problems for the Sturm–Liouville operator with discontinuity inside a finite interval was studied. In fact, it is unnecessary to fix above mentioned parameters. In this work, we prove that the potential function and all parameters in discontinuity conditions and boundary conditions can be uniquely determined by a set of values of eigenfunctions in some interior point and one spectrum. We also establish Gesztesy–Simon type theorem and Ambarzumyan type theorem of inverse spectral aspects for discontinuous boundary value problem. The proofs in this work rely on the properties of analytic functions and Phragmén-Lindelöf theorem.