arXiv subjects

Elyor B. Dilmurodov

Publications and source records attributed to Elyor B. Dilmurodov.

4 recordsLinked to original sources

arXiv2020

Discrete Eigenvalues of a $2 \times 2$ Operator Matrix

We consider a $2\times2$ block operator matrix ${\mathcal A}_μ$ $($$μ>0$ is a coupling constant$)$ acting in the direct sum of one- and two-particle subspaces of a bosonic Fock space. The location of the essential spectrum of ${\mathcal A}_μ$ is described and its bounds are estimated. It is shown that there exist the critical values $μ_l^0(γ)$ with $γ>0$ and $μ_r^0(γ)$ with $γ<12$ of the coupling constant $μ>0$ such that for all $γ>0$ $(γ<12)$ the operator ${\mathcal A}_μ$ with $μ=μ_l^0(γ)$ $(μ=μ_r^0(γ)$ has infinitely many eigenvalues on the l.h.s. $($r.h.s.$)$ of the its essential spectrum. We prove that for all $μ\not\in \{μ_l^0(γ),μ_r^0(γ)\}$ the operator ${\mathcal A}_μ$ has finitely many discrete eigenvalues on the l.h.s. and r.h.s. of its essential spectrum.

math.FA↗

arXiv2020

Analysis of the spectrum of a $2\times 2$ operator matrix. Discrete spectrum asymptotics

We consider a $2 \times 2$ operator matrix ${\mathcal A}_μ,$ $μ>0$ related with the lattice systems describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles. We obtain an analogue of the Faddeev equation and its symmetric version for the eigenfunctions of ${\mathcal A}_μ$. We describe the new branches of the essential spectrum of ${\mathcal A}_μ$ via the spectrum of a family of generalized Friedrichs models. It is established that the essential spectrum of ${\mathcal A}_μ$ consists the union of at most three bounded closed intervals and their location is studied. For the critical value $μ_0$ of the coupling constant $μ$ we establish the existence of infinitely many eigenvalues, which are located in the both sides of the essential spectrum of ${\mathcal A}_μ$. In this case, an asymptotic formula for the discrete spectrum of ${\mathcal A}_μ$ is found.

math-ph↗

arXiv2019

Threshold analysis for a family of $2 \times 2$ operator matrices

We consider a family of $2 \times 2$ operator matrices ${\mathcal A}_μ(k),$ $k \in {\Bbb T}^3:=(-π, π]^3,$ $μ>0$, 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 ${\Bbb Z}^3,$ interacting via annihilation and creation operators. We find a set $Λ:=\{k^{(1)},...,k^{(8)}\} \subset {\Bbb T}^3$ and a critical value of the coupling constant $μ$ to establish necessary and sufficient conditions for either $z=0=\min\limits_{k\in {\Bbb T}^3} σ_{\rm ess}({\mathcal A}_μ(k))$ ( or $z=27/2=\max\limits_{k\in {\Bbb T}^3} σ_{\rm ess}({\mathcal A}_μ(k))$ is a threshold eigenvalue or a virtual level of ${\mathcal A}_μ(k^{(i)})$ for some $k^{(i)} \in Λ.$

math-ph↗

arXiv2019

Eigenvalues and virtual levels of a family of $2 \times 2$ operator matrices

In the present paper we consider a family of $2 \times 2$ operator matrices ${\mathcal A}_μ(k),$ $k \in {\Bbb T}^3:=(-π, π]^3,$ $μ>0,$ associated with the Hamiltonian of a system consisting of at most two particles on a three-dimensional lattice ${\Bbb Z}^3,$ interacting via creation and annihilation operators. We prove that there is a value $μ_0$ of the parameter $μ$ such that only for $μ=μ_0$ the operator ${\mathcal A}_μ(\overline{0})$ has a virtual level at the point $z=0=\minσ_{\rm ess}({\mathcal A}_μ(\overline{0}))$ and the operator ${\mathcal A}_μ(\overlineπ)$ has a virtual level at the point $z=18=\maxσ_{\rm ess}({\mathcal A}_μ(\overlineπ))$, where $\overline{0}:=(0,0,0), \overlineπ:=(π,π,π) \in {\Bbb T}^3.$ The absence of the eigenvalues of ${\mathcal A}_μ(k)$ for all values of $k$ under the assumption that $μ=μ_0$ is shown. The threshold energy expansions for the Fredholm determinant associated to ${\mathcal A}_μ(k)$ are obtained.

math-ph↗