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This article is cited in 5 scientific papers (total in 5 papers)
MATHEMATICS
Threshold analysis for a family of $2\times2$ operator matrices
T. H. Rasulov, E. B. Dilmurodov Department of Mathematics, Faculty of Physics and Mathematics,
Bukhara State University, M. Ikbol str. 11, 200100 Bukhara, Uzbekistan
Abstract:
We consider a family of $2\times2$ operator matrices $\mathcal{A}_\mu(k)$, $k\in\mathbb{T}^3:=(-\pi;\pi]^3$, $\mu>0$, acting in the direct sum of zero- and one-particle subspaces of a Fock space. It is associated with the Hamiltonian of a system consisting of at most two particles on a three-dimensional lattice $\mathbb{Z}^3$, interacting via annihilation and creation operators. We find a set $\Lambda:=\{k^{(1)},\dots,k^{(8)}\}\subset\mathbb{T}^3$ and a critical value of the coupling constant $\mu$ to establish necessary and sufficient conditions for either $z=0=\min\limits_{k\in\mathbb{T}^3}\sigma_{\mathrm{ess}}(\mathcal{A}_\mu(k))$ (or $z=27/2=\max\limits_{k\in\mathbb{T}^3}\sigma_{\mathrm{ess}}(\mathcal{A}_\mu(k))$) is a threshold eigenvalue or a virtual level of $\mathcal{A}_\mu(k^{(i)})$ for some $k^{(i)}\in\Lambda$.
Keywords:
operator matrices, Hamiltonian, generalized Friedrichs model, zero- and one-particle subspaces of a Fock space, threshold eigenvalues, virtual levels, annihilation and creation operators.
Received: 19.10.2019 Revised: 13.11.2019
Citation:
T. H. Rasulov, E. B. Dilmurodov, “Threshold analysis for a family of $2\times2$ operator matrices”, Nanosystems: Physics, Chemistry, Mathematics, 10:6 (2019), 616–622
Linking options:
https://www.mathnet.ru/eng/nano475 https://www.mathnet.ru/eng/nano/v10/i6/p616
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