Abstract
AbstractThe inverse spectral problem is studied for the Sturm-Liouville operator with a complex-valued potential and arbitrary entire functions in one of the boundary conditions. We obtain necessary and sufficient conditions for uniqueness and develop a constructive algorithm for the inverse problem solution. The main results are applied to the Hochstadt-Lieberman half-inverse problem. As an auxiliary proposition, we prove local solvability and stability for the inverse Sturm-Liouville problem by the Cauchy data in the non-self-adjoint case.
Highlights
IntroductionThe article aims to solve the inverse spectral problem for the following boundary value problem L (q):
The article aims to solve the inverse spectral problem for the following boundary value problem L (q):− y′′(x) + q (x)y (x) = λy (x), x ∈ (0, π), (1.1)y (0) = 0, f1 (λ)y′(π) + f2 (λ)y (π) = 0. (1.2)Here (1.1) is the Sturm-Liouville equation with the complex-valued potential q ∈ L2 (0, π)
For solving the inverse Sturm-Liouville problem with boundary conditions independent of the spectral parameter, we rely on the inverse problem theory for non-self-adjoint SturmLiouville operators developed in [4,38,39]
Summary
The article aims to solve the inverse spectral problem for the following boundary value problem L (q):. The authors of [35] considered Gestezy-Simon-type inverse problems [36] for the Sturm-Liouville equation (1.1) with the potential given on (a, π), a ∈ (0, π), and represented those problems in the form with the boundary condition y′(a) − f (λ)y (a) = 0, where f (λ) is a Herglotz function. Our method is based on completeness and basisness of special vector-functional sequences in appropriate Hilbert spaces This method allows us to reduce Inverse Problem 1.1 to the classical SturmLiouville inverse problem with constant coefficients in the boundary conditions. For solving the inverse Sturm-Liouville problem with boundary conditions independent of the spectral parameter, we rely on the inverse problem theory for non-self-adjoint SturmLiouville operators developed in [4,38,39]. As far as we know, Theorem 6.1 is new for the case of the complex-valued potential q (x) and so can be treated as a separate result
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
