Spectral problems of nonself-adjoint singular discrete Sturm-Liouville operators

Year, pages: 2016, 967 - 978

In this study we construct a space of boundary values of the minimal symmetric discrete Sturm-Liouville (or second-order difference) operators with defect index $(1,1)$ (in limit-circle case at $\pm ∞$ and limit-point case at $\mp ∞$), acting in the Hilbert space $\ell_ρ²(\mathbb{Z})$ ($\mathbb{Z} :=\{0,\pm 1,\pm 2,…\}$). Such a description of all maximal dissipative, maximal accumulative and self-adjoint extensions is given in terms of boundary conditions at $\pm∞$. After constructing the space of the boundary values, we investigate two classes of maximal dissipative operators. This investigation is done with the help of the boundary conditions, called ``dissipative at $-∞$'' and ``dissipative at $∞$''. In each of these cases we construct a self-adjoint dilation of maximal dissipative operator and its incoming and outgoing spectral representations. These representations allow us to determine the scattering matrix of dilation. We also construct a functional model of the maximal dissipative operator and define its characteristic function in terms of the Weyl-Titchmarsh function of the self-adjoint operator. Finally, we prove a theorem on completeness of the system of eigenvectors and associated vectors (or root vectors) of the maximal dissipative operators.

Allahverdiev, B. 2016. Spectral problems of nonself-adjoint singular discrete Sturm-Liouville operators. In Mathematica Slovaca, vol. 66, no.4, pp. 967-978. 0139-9918. DOI: https://doi.org/10.1515/ms-2015-0196

Allahverdiev, B. (2016). Spectral problems of nonself-adjoint singular discrete Sturm-Liouville operators. Mathematica Slovaca, 66(4), 967-978. 0139-9918. DOI: https://doi.org/10.1515/ms-2015-0196