Allahverdiev, Bilender P.2021-10-132021-10-132019-01-02Allahverdiev, B. P. (2019). Spectral analysis of singular Hamiltonian systems with an eigenparameter in the boundary condition. <i>Electronic Journal of Differential Equations, 2019</i>(02), pp. 1-14.1072-6691https://hdl.handle.net/10877/14645In this article we study a non-self-adjoint eigenparameter dependent singular differential 1D Hamiltonian system with the singular end points a and b in the Hilbert space L2P((a,b);\ ℂ2 and we consider that this 1D Hamiltonian system is in the limit-circle cases at a and b. For this purpose we use the maximal dissipative operator associated with the considered problem whose spectral analysis is sufficient for boundary value problem. Self-adjoint dilation theory of Sz.-Nagy-Foias developed for the dissipative operators is used. Moreover we construct incoming and outgoing spectral representations of the self-adjoint dilation. This representations allows us to determine the scattering matrix. Therefore a functional model of the dissipative operator is constructed. Moreover, a functional model of the dissipative operator is constructed and its characteristic function in terms of solutions of the corresponding Hamiltonian system is described. Therefore using the obtained results for the characteristic function theory, theorems on completeness of the system of eigenvectors and associated vectors of the dissipative operator and Hamiltonian boundary value problem have been proved.Text14 pages1 file (.pdf)enAttribution 4.0 InternationalHamiltonian systemLimit-circleDissipative operatorSpectral parameter in the boundarySelf-adjoint dilationScattering matrixFunctional modelCharacteristic functionArticleThis work is licensed under a Creative Commons Attribution 4.0 International License.