arXiv subjects

Tulkin H. Rasulov

Publications and source records attributed to Tulkin H. Rasulov.

11 recordsLinked to original sources

arXiv2020

Analysis of the discrete spectrum of the family of $3 \times 3$ operator matrices

We consider the family of $3 \times 3$ operator matrices ${\bf H}(K),$ $K \in {\Bbb T}^3:=(-π; π]^3$ associated with the lattice systems describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles. We find a finite set $Λ\subset {\Bbb T}^3$ to prove the existence of infinitely many eigenvalues of ${\bf H}(K)$ for all $K \in Λ$ when the associated Friedrichs model has a zero energy resonance. It is found that for every $K \in Λ,$ the number $N(K, z)$ of eigenvalues of ${\bf H}(K)$ lying on the left of $z,$ $z<0,$ satisfies the asymptotic relation $\lim\limits_{z \to -0} N(K, z) |\log|z||^{-1}={\mathcal U}_0$ with $0<{\mathcal U}_0<\infty,$ independently on the cardinality of $Λ.$ Moreover, we prove that for any $K \in Λ$ the operator ${\bf H}(K)$ has a finite number of negative eigenvalues if the associated Friedrichs model has a zero eigenvalue or a zero is the regular type point for positive definite Friedrichs model.

math-ph↗

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

Analytic description of the essential spectrum of a family of $3 \times 3$ operator matrices

We consider the family of $3 \times 3$ operator matrices $H(K),$ $K \in {\Bbb T}^{\rm d}:=(-π; π]^{\rm d}$ arising in the spectral analysis of the energy operator of the spin-boson model of radioactive decay with two bosons on the torus ${\Bbb T}^{\rm d}.$ We obtain an analogue of the Faddeev equation for the eigenfunctions of $H(K)$. An analytic description of the essential spectrum of $H(K)$ is established. Further, it is shown that the essential spectrum of $H(K)$ consists the union of at most three bounded closed intervals.

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↗

arXiv2016

On the finiteness of the discrete spectrum of a 3x3 operator matrix

An operator matrix $H$ associated with a lattice system describing three particles in interactions, without conservation of the number of particles, is considered. The structure of the essential spectrum of $H$ is described by the spectra of two families of the generalized Friedrichs models. A symmetric version of the Weinberg equation for eigenvectors of $H$ is obtained. The conditions which guarantee the finiteness of the number of discrete eigenvalues located below the bottom of the three-particle branch of the essential spectrum of $H$ is found.

math.SP↗

arXiv2010

The Faddeev Equation and Essential Spectrum of a Hamiltonian in Fock Space

A Hamiltonian (model operator) $H$ associated to a quantum system describing three particles in interaction, without conservation of the number of particles, is considered. The Faddeev type system of equations for eigenvectors of $H$ is constructed. The essential spectrum of $H$ is described by the spectrum of the channel operator.

math-ph↗

arXiv2009

Discrete Spectrum of a Model Operator Related to Three-Particle Discrete Schrödinger Operators

A model operator $H_μ,$ $μ>0$ associated to a system of three particles on the three-dimensional lattice $ \mathbb{Z}^3$ that interact via nonlocal pair potentials is considered. We study the case where the parameter function $w$ has a special form with the non degenerate minimum at the $n, n>1$ points of the six-dimensional torus $\mathbb{T}^6.$ If the associated Friedrichs model has a zero energy resonance, then we prove that the operator $H_μ$ has infinitely many negative eigenvalues accumulating at zero and we obtain an asymptotics for the number of eigenvalues of $H_μ$ lying below $z,$ $z<0$ as $z\to -0.$

math.FA↗

arXiv2008

On the Spectrum of a Model Operator in Fock Space

A model operator $H$ associated to a system describing four particles in interaction, without conservation of the number of particles, is considered. We describe the essential spectrum of $H$ by the spectrum of the channel operators and prove the Hunziker-van Winter-Zhislin (HWZ) theorem for the operator $H.$ We also give some variational principles for boundaries of the essential spectrum and interior eigenvalues.

math-ph↗

arXiv2005

On the spectrum of an Hamiltonian in Fock space. Discrete spectrum Asymptotics

A model operator $H$ associated with the energy operator of a system describing three particles in interaction, without conservation of the number of particles, is considered. The precise location and structure of the essential spectrum of $H$ is described. The existence of infinitely many eigenvalues below the bottom of the essential spectrum of $H$ is proved for the case where an associated generalized Friedrichs model has a resonance at the bottom of its essential spectrum. An asymptotics for the number $N(z)$ of eigenvalues below the bottom of the essential spectrum is also established. The finiteness of eigenvalues of $H$ below the bottom of the essential spectrum is proved if the associated generalized Friedrichs model has an eigenvalue with energy at the bottom of its essential spectrum.

math-ph↗

arXiv2005

The number of eigenvalues for an Hamiltonian in Fock space

A model operator $H$ corresponding to the energy operator of a system with non-conserved number $n\leq 3$ of particles is considered. The precise location and structure of the essential spectrum of $H$ is described. The existence of infinitely many eigenvalues below the bottom of the essential spectrum of $H$ is proved if the generalized Friedrichs model has a virtual level at the bottom of the essential spectrum and for the number $N(z)$ of eigenvalues below $z<0$ an asymptotics established. The finiteness of eigenvalues of $H$ below the bottom of the essential spectrum is proved if the generalized Friedrichs model has a zero eigenvalue at the bottom of its essential spectrum.

math-ph↗