Spectral analysis of singular Hamiltonian systems with an eigenparameter in the boundary condition

• Main Author: Bilender P. Allahverdiev
• Format: Article
• Language: English
• Published: Texas State University 2019-01-01
• Series: Electronic Journal of Differential Equations
• Subjects: Hamiltonian system; limit-circle; dissipative operator; spectral parameter in the boundary; self-adjoint dilation,; scattering matrix; functional model; characteristic function
• Online Access: http://ejde.math.txstate.edu/Volumes/2019/02/abstr.html

In 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 $L_P^2((a,b);\mathbb{C}^2)$ 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.