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# Eigenvalues and virtual levels of a family of 2×2 operator matrices

### Abstract

In the present paper we consider a family of $2 \times 2$ operator matrices ${\mathcal A}_\mu(k),$ $k \in {\mathbb T}^3:=(-\pi, \pi]^3,$ $\mu>0,$ associated with the Hamiltonian of a system consisting of at most two particles on a three-dimensional lattice ${\mathbb Z}^3,$ interacting via creation and annihilation operators. We prove that there is a value $\mu_0$ of the parameter $\mu$ such that only for $\mu=\mu_0$ the operator ${\mathcal A}_\mu(\overline{0})$ has a virtual level at the point $z=0=\min\sigma_{\rm ess}({\mathcal A}_\mu(\overline{0}))$ and the operator ${\mathcal A}_\mu(\overline{\pi})$ has a virtual level at the point $z=18=\max\sigma_{\rm ess}({\mathcal A}_\mu(\overline{\pi}))$, where $\overline{0}:=(0,0,0), \overline{\pi}:=(\pi,\pi,\pi) \in {\mathbb T}^3.$ The absence of the eigenvalues of ${\mathcal A}_\mu(k)$ for all values of $k$ under the assumption that $\mu=\mu_0$ is shown. The threshold energy expansions for the Fredholm determinant associated to ${\mathcal A}_\mu(k)$ are obtained.

Key words: Operator matrices, eigenvalues, virtual levels, creation and annihilation operators, Fredholm determinant.

### Article Information

 Title Eigenvalues and virtual levels of a family of 2×2 operator matrices Source Methods Funct. Anal. Topology, Vol. 25 (2019), no. 3, 273-281 MathSciNet MR4016213 Milestones Received 11/03/2019 Copyright The Author(s) 2019 (CC BY-SA)

### Authors Information

Tulkin H. Rasulov
Department of Mathematics, Faculty of Physics and Mathematics, Bukhara State University, M. Ikbol str. 11, 200100 Bukhara, Uzbekistan

Elyor B. Dilmurodov
Department of Mathematics, Faculty of Physics and Mathematics, Bukhara State University, M. Ikbol str. 11, 200100 Bukhara, Uzbekistan

### Citation Example

Tulkin H. Rasulov and Elyor B. Dilmurodov, Eigenvalues and virtual levels of a family of 2×2 operator matrices, Methods Funct. Anal. Topology 25 (2019), no. 3, 273-281.

### BibTex

@article {MFAT1211,
AUTHOR = {Tulkin H. Rasulov and Elyor B. Dilmurodov},
TITLE = {Eigenvalues and virtual levels of a family of 2×2 operator matrices},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {25},
YEAR = {2019},
NUMBER = {3},
PAGES = {273-281},
ISSN = {1029-3531},
MRNUMBER = {MR4016213},
URL = {http://mfat.imath.kiev.ua/article/?id=1211},
}

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