Taiwanese Journal of Mathematics

Functional Model and Spectral Analysis of Discrete Singular Hamiltonian System

Bilender P. Allahverdiev

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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.

Article information

Taiwanese J. Math., Volume 23, Number 3 (2019), 653-673.

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

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Zentralblatt MATH identifier

Primary: 47B39: Difference operators [See also 39A70] 47B44: Accretive operators, dissipative operators, etc. 47A20: Dilations, extensions, compressions 47A40: Scattering theory [See also 34L25, 35P25, 37K15, 58J50, 81Uxx] 47A45: Canonical models for contractions and nonselfadjoint operators
Secondary: 47B25: Symmetric and selfadjoint operators (unbounded) 47A75: Eigenvalue problems [See also 47J10, 49R05] 39A70: Difference operators [See also 47B39]

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


Allahverdiev, Bilender P. Functional Model and Spectral Analysis of Discrete Singular Hamiltonian System. Taiwanese J. Math. 23 (2019), no. 3, 653--673. doi:10.11650/tjm/181007. https://projecteuclid.org/euclid.twjm/1540195381

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