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June, 2019 Functional Model and Spectral Analysis of Discrete Singular Hamiltonian System
Bilender P. Allahverdiev
Taiwanese J. Math. 23(3): 653-673 (June, 2019). DOI: 10.11650/tjm/181007


A space of boundary values is constructed for a minimal symmetric operator, generated by a discrete singular Hamiltonian system, acting in the Hilbert space $\ell_{\mathbf{A}}^{2}(\mathbb{N}_{0}; E \oplus E)$ ($\mathbb{N}_{0} = \{ 0,1,2,\ldots \}$, $\dim E = m \lt \infty$) with maximal deficiency indices $(m,m)$ (in limit-circle case). A description of all maximal dissipative, maximal accumulative, self-adjoint and other extensions of such a symmetric operator is given in terms of boundary conditions at infinity. We construct a self-adjoint dilation of a maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We establish a functional model of the dissipative operator and construct its characteristic function in terms of the scattering matrix of the dilation. Finally, we prove the theorem on completeness of the system of eigenvectors and associated vectors (or root vectors) of the dissipative operator.


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Bilender P. Allahverdiev. "Functional Model and Spectral Analysis of Discrete Singular Hamiltonian System." Taiwanese J. Math. 23 (3) 653 - 673, June, 2019.


Received: 22 January 2018; Revised: 6 June 2018; Accepted: 8 October 2018; Published: June, 2019
First available in Project Euclid: 22 October 2018

zbMATH: 07068568
MathSciNet: MR3952245
Digital Object Identifier: 10.11650/tjm/181007

Primary: 47A20 , 47A40 , 47A45 , 47B39 , 47B44
Secondary: 39A70 , 47A75 , 47B25

Keywords: Characteristic function , completeness of the root vectors , deficiency indices , discrete Hamiltonian system , functional model , minimal symmetric operator , scattering matrix , self-adjoint and maximal dissipative extensions of minimal operator , self-adjoint dilation , space of boundary values

Rights: Copyright © 2019 The Mathematical Society of the Republic of China


Vol.23 • No. 3 • June, 2019
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