arXiv subjects

Saidakhmat N. Lakaev

Publications and source records attributed to Saidakhmat N. Lakaev.

At least 19 recordsLinked to original sources

arXiv2026

Threshold Virtual States of a Jacobi operator

We prove that the set of parameters for which a virtual level appears at the edge of the continuous spectrum of a Jacobi matrix with a finite-rank diagonal perturbation constitutes an algebraic variety of codimension one. This variety partitions the parameter space into connected components, with their number determined by the size of the perturbation support. We also reveal a hierarchical structure underlying these critical varieties as the rank of the perturbation increases.

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arXiv2025

Threshold Resonances, Critical Couplings, and Eigenvalue Bounds for Two-Particle Operators on $\mathbb{Z}^3$

We study a family of lattice Schrödinger operators $H_{μ_1μ_2}(K)$ describing two identical bosons on the three-dimensional cubic lattice $\mathbb{Z}^3$, where $K \in \mathbb{T}^3$ is the quasi-momentum, and $μ_1, μ_2 \in \mathbb{R}$ are coupling constants corresponding to on-site and nearest-neighbour interactions, respectively. We show that the Hilbert space $L^{2,\mathrm{e}}(\mathbb{T}^3)$ decomposes into three mutually orthogonal subspaces, each invariant under $H_{μ_1μ_2}(0)$. A detailed spectral analysis of the restriction of $H_{μ_1μ_2}(0)$ to one of these subspaces reveals two smooth critical curves in the $(μ_1, μ_2)$-plane, separating regions where the number of eigenvalues below the essential spectrum remains constant. For the restrictions to the other two subspaces, we identify a critical point on the $μ_2$-axis that partitions it into intervals with a constant number of eigenvalues below the essential spectrum. Analogously, two additional critical curves and one critical point determine regions and intervals where the number of eigenvalues above the essential spectrum is constant. In particular, for suitable parameter values, $H_{μ_1μ_2}(0)$ may possess up to three bound states located either below or above the essential spectrum, with numbers $(α,β)$ satisfying $α+ β< 3$ or $(α,β) \in \{(3,0),(0,3)\}$. Here, by \emph{eigenvalue bounds} we mean both the possible locations of eigenvalues outside the essential spectrum and the maximum number of such eigenvalues for given coupling parameters. Finally, we extend the analysis to arbitrary quasi-momentum $K \in \mathbb{T}^3$, obtaining general lower bounds for the number of eigenvalues of $H_{μ_1μ_2}(K)$.

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arXiv2024

Number of bound states of the Hamiltonian of a lattice two-boson system with interactions up to the next neighbouring sites

We study the family $H_{γλμ}(K)$, $K\in \mathbb{T}^2,$ of discrete Schrödinger operators, associated to the Hamiltonian of a system of two identical bosons on the two-dimen\-sional lattice $\mathbb{Z}^2,$ interacting through on one site, nearest-neighbour sites and next-nearest-neighbour sites with interaction magnitudes $γ,λ$ and $μ,$ respectively. We prove there existence an important invariant subspace of operator $H_{γλμ}(0)$ such that the restriction of the operator $H_{γλμ}(0)$ on this subspace has at most two eigenvalues lying both as below the essential spectrum as well as above it, depending on the interaction magnitude $λ,μ\in \mathbb{R}$ (only). We also give a sharp lower bound for the number of eigenvalues of $H_{γλμ}(K)$.

math-ph↗

arXiv2023

Two-fermion lattice Hamiltonian with first and second nearest-neighboring-site interactions

We study the Schroedinger operators H_{λμ}(K), with K \in T_2 the fixed quasi-momentum of the particles pair, associated with a system of two identical fermions on the two-dimensional lattice Z_2 with first and second nearest-neighboring-site interactions of magnitudes λ\in R and μ\in R, respectively. We establish a partition of the (λ,μ)-plane so that in each its connected component, the Schroedinger operator H_{λμ}(0) has a definite (fixed) number of eigenvalues, which are situated below the bottom of the essential spectrum and above its top. Moreover, we establish a sharp lower bound for the number of isolated eigenvalues of H_{λμ}(K) in each connected component.

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arXiv2023

The number and location of eigenvalues for the two-particle Schrödinger operators on lattices

We study the Schrödinger operators $H_{γλμ}(K)$, $K\in\T$ being a fixed (quasi)momentum of the particles pair, associated with a system of two identical bosons on the one-dimensional lattice $\mathbb{Z}$, where the real quantities $γ$, $λ$ and $μ$ describe the interactions between pairs of particles on one site, two nearest neighboring sites and next two neighboring sites, respectively. We found a partition of the three-dimensional space $(γ, λ,μ)$ of interaction parameters into connected components and the exact number of eigenvalues of this operator that lie below and above the essential spectrum, in each component. Moreover, we show that for any $K\in\T^d$ the number of eigenvalues of $H_{γλμ}(K)$ is not less than the corresponding number of eigenvalues of $H_{γλμ}(0)$.

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arXiv2022

On the spectrum of Schrödinger-type operators on two dimensional lattices

We consider a family $$ \widehat H_{a,b}(μ)=\widehat H_0 +μ\widehat V_{a,b}\quad μ>0, $$ of Schrödinger-type operators on the two dimensional lattice $\mathbb{Z}^2,$ where $\widehat H_0$ is a Laurent-Toeplitz-type convolution operator with a given Hopping matrix $\hat{e}$ and $\widehat V_{a,b}$ is a potential taking into account only the zero-range and one-range interactions, i.e., a multiplication operator by a function $\hat v$ such that $\hat v(0)=a,$ $\hat v(x)=b$ for $|x|=1$ and $\hat v(x)=0$ for $|x|\ge2,$ where $a,b\in\mathbb{R}\setminus\{0\}.$ Under certain conditions on the regularity of $\hat{e}$ we completely describe the discrete spectrum of $\hat H_{a,b}(μ)$ lying above the essential spectrum and study the dependence of eigenvalues on parameters $μ,$ $a$ and $b.$ Moreover, we characterize the threshold eigenfunctions and resonances.

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arXiv2020

Threshold effects of the two-particle Schrödinger operators on lattices

We consider a wide class of the two-particle Schrödinger operators $H_μ(k)=H_{0}(k)+μV, \,μ>0,$ with a fixed two-particle quasi-momentum $k$ in the $d$ -dimensional torus $\mathbb{T}^d$, associated to the Bose-Hubbard hamiltonian $H_μ$ of a system of two identical quantum-mechanical particles (bosons) on the $d$- dimensional hypercubic lattice $\mathbb{Z}% ^d$ interacting via short-range pair potentials. We study the existence of eigenvalues of $H_μ(k)$ below the threshold of the essential spectrum depending on the interaction energy $μ>0$ and the quasi-momentum $k\in \mathbb{T}^d$ of particles. We prove that the threshold (bottom of the essential spectrum), as a singular point (a threshold resonance or a threshold eigenvalue), creates eigenvalues below the essential spectrum under perturbations of both the coupling constant $μ>0$ and the quasi-momentum $k$ of the particles. Moreover, we show that if the threshold is a regular point, then it does not create any eigenvalues under small perturbations of the coupling constant $μ>0$ and the quasi-momentum $k$.

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arXiv2016

The existence of bound states in a system of three particles in an optical lattice

We consider the hamiltonian $\mathrm{H}_μ,μ\in \R$ of a system of three-particles (two identical fermions and one different particle) moving on the lattice ${\Z}^d ,\, d=1,2 $ interacting through repulsive $(μ>0)$ or attractive $(μ<0)$ zero-range pairwise potential $μV$. We prove for any $μ\ne0$ the existence of bound state of the discrete three-particle Schrödinger operator $H_μ(K),\,K\in \T^d$ being the three-particle quasi-momentum, associated to the hamiltonian $\mathrm{H}_μ$.

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arXiv2015

The existence of an isolated band in a system of three particles in an optical lattice

We prove the existence of two-and three-particle bound states of the Schrödinger operators $h_μ(k),k\in \T^d$ and $H_μ(K),K\in \T^d$ associated to Hamiltonians $\mathrm{h}_μ$ and $\mathrm{H}_μ$ of a system of two and three identical bosons on the lattice $\Z^d, d=1,2$ interacting via pairwise zero-range attractive $μ<0$ or repulsive $μ>0$ potentials. As a consequence we show the existence of an isolated band in the two and three bosonic systems in an optical lattice.

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arXiv2015

Number of bound states of the Schroedinger operator of a system of three bosons in an optical lattice

We consider the Hamiltonian $\hat {\mathrm{H}}_μ$ of a system of three identical particles(bosons) on the $d-$ dimensional lattice $\Z^d, d=1,2$ interacting via pairwise zero-range attractive potential $μ<0$. We describe precise location and structure of the essential spectrum of the Schrödinger operator $H_μ(K),K\in \T^d$ associated to $\hat {\mathrm{H}}_μ$ and prove the finiteness of the number of bound states of $H_μ(K),K\in \T^d$ lying below the bottom of the essential spectrum. Moreover, we show that bound states decay exponentially at infinity and eigenvalues and corresponding bound states of $H_μ(K),K\in \T^d$ are regular as a function of center of mass quasi-momentum $K\in \T^d$.

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arXiv2015

The existence and location of eigenvalues of the one particle discrete Schroedinger operators

We consider a quantum particle moving in the one dimensional lattice Z and interacting with a indefinite sign external field v. We prove that the associated discrete Schroedinger operator H can have one or two eigenvalues, situated as below the bottom of the essential spectrum, as well as above its top. Moreover, we show that the operator H can have two eigenvalues outside of the essential spectrum such that one of them is situated below the bottom of the essential spectrum, and other one above its top.

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arXiv2014

Threshold phenomenon for a family of the Generalized Generalized Friedrichs models with the perturbation of rank one

A family $H_μ(p),$ $μ>0,$ $p\in\T^3$ of the Generalized Firedrichs models with the perturbation of rank one, associated to a system of two particles, moving on the three dimensional lattice $\mathbb{\Z}^3,$ is considered. The existence or absence of the unique eigenvalue of the operator $H_μ(p)$ lying outside the essential spectrum, depending on the values of $μ>0$ and $p\in U_δ(p_{\,0})\subset\T^3$ is proven. Moreover, the analyticity of associated eigenfunction is shown.

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arXiv2010

Asymptotics of Eigenvalues of the Two-particle Schrödinger operators on lattices

The Hamiltonian of a system of two quantum mechanical particles moving on the $d$-dimensional lattice $\Z^d$ and interacting via zero-range attractive pair potentials is considered. For the two-particle energy operator $H_μ(K),$ $K\in \T^d=(-π,π]^d$ -- the two-particle quasi-momentum, the existence of a unique positive eigenvalue $z(μ, K)$ above the upper edge of the essential spectrum of $H_μ(K)$ is proven and asymptotics for $z(μ, K)$ are found when $μ$ approaches to some $μ_0(K)$ and $K\to 0.$

math-ph↗

arXiv2007

Discrete spectrum asymptotics for the three-particle Hamiltonians on lattices

We consider the Hamiltonian of a system of three quantum mechanical particles on the three-dimensional lattice $\Z^3$ interacting via short-range pair potentials. We prove for the two-particle energy operator $h(k),$ $k\in \T^3$ the two-particle quasi-momentum, the existence of a unique positive eigenvalue $z(k)$ lying below the essential spectrum under assumption that the operator $h(0)$ corresponding to the zero value of $k$ has a zero energy resonance. We describe the location of the essential spectrum of the three-particle discrete Schrödinger operators $H(K)$,$K$ the three-particle quasi-momentum by the spectra of $h(k), k\in \T^3.$ We prove the existence of infinitely many eigenvalues of H(0) and establish for the number of eigenvalues $N(0,z)$ lying below $z<0$ the asymptotics \begin{equation*}\label{asimz} \lim\limits_{z \to -0}\frac{N(0,z)}{|\log |z||}=\frac{λ_0}{2π}, \end{equation*} where $λ_0$ a unique positive solution of the equation $$ λ= \frac{8 \sinh πλ/6}{\sqrt 3 \cosh πλ/2}.$$ We prove that for all $ K \in U_δ^0(0),$ where $U_δ^0(0)$ some punctured $δ>0$ neighborhood of the origin, the number $N(K,0)$ of eigenvalues the operator $H(K)$ below zero is finite and satisfy the asymptotics \begin{equation*}\label{asimk} \lim\limits_{|K| \to 0}\frac{N(K,0)}{|\log |K||}=\frac{λ_0}π. \end{equation*}

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arXiv2007

Asymptotics for the number of eigenvalues of three-particle Schrödinger operators on lattices

We consider the Hamiltonian of a system of three quantum mechanical particles (two identical fermions and boson)on the three-dimensional lattice $\Z^3$ and interacting by means of zero-range attractive potentials. We describe the location and structure of the essential spectrum of the three-particle discrete Schrödinger operator $H_γ(K),$ $K$ being the total quasi-momentum and $γ>0$ the ratio of the mass of fermion and boson. We choose for $γ>0$ the interaction $v(γ)$ in such a way the system consisting of one fermion and one boson has a zero energy resonance. We prove for any $γ> 0$ the existence infinitely many eigenvalues of the operator $H_γ(0).$ We establish for the number $N(0,γ; z;)$ of eigenvalues lying below $z<0$ the following asymptotics $$ \lim_{z\to 0-}\frac{N(0,γ;z)}{\mid \log \mid z\mid \mid}={U} (γ) .$$ Moreover, for all nonzero values of the quasi-momentum $K \in T^3 $ we establish the finiteness of the number $ N(K,γ;τ_{ess}(K))$ of eigenvalues of $H(K)$ below the bottom of the essential spectrum and we give an asymptotics for the number $N(K,γ;0)$ of eigenvalues below zero.

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arXiv2006

On the number of eigenvalues of a model operator associated to a system of three-particles on lattices

A model operator $H$ associated to a system of three-particles on the three dimensional lattice $\Z^3$ and interacting via pair non-local potentials is studied. The following results are proven: (i) the operator $H$ has infinitely many eigenvalues lying below the bottom of the essential spectrum and accumulating at this point, in the case, where both Friedrichs model operators $h_{μ_α}(0),α=1,2,$ have threshold resonances. (ii) the operator $H$ has a finite number of eigenvalues lying outside of the essential spectrum, in the case, where at least one of $h_{μ_α}(0), α=1,2,$ has a threshold eigenvalue.

math-ph↗

arXiv2006

The threshold effects for a family of Friedrichs models under rank one perturbations

A family of Friedrichs models under rank one perturbations $h_μ(p),$ $p \in (-π,π]^3$, $μ>0,$ associated to a system of two particles on the three dimensional lattice $\Z^3$ is considered. We prove the existence of a unique eigenvalue below the bottom of the essential spectrum of $h_μ(p)$ for all nontrivial values of $p$ under the assumption that $h_μ(0)$ has either a threshold energy resonance (virtual level) or a threshold eigenvalue. The threshold energy expansion for the Fredholm determinant associated to a family of Friedrichs models is also obtained.

math.SP↗

arXiv2006

Low energy effects for a family of Friedrichs models

A family of Friedrichs models with rank one perturbations $h_μ(p),$ $p \in (-π,π]^3,μ>0$ associated to a system of two particles on the lattice $\Z^3$ is considered. The existence of a unique strictly positive eigenvalue below the bottom of the essential spectrum of $h_μ(p)$ for all nontrivial values $p \in (-π,π]^3$ under the assumption that $h_μ(0)$ has either a zero energy resonance (virtual level) or a threshold eigenvalue is proved. Low energy asymptotic expansion for the Fredholm determinant associated to family of Friedrichs models is obtained.

math.SP↗