arXiv subjects

Saidakhmat Lakaev

Publications and source records attributed to Saidakhmat Lakaev.

5 recordsLinked to original sources

arXiv2021

Bose-Hubbard models with on-site and nearest-neighbor interactions: Exactly solvable case

We study the discrete spectrum of the two-particle Schrödinger operator $\hat H_{μλ}(K),$ $K\in\mathbb{T}^2,$ associated to the Bose-Hubbard Hamiltonian $\hat {\mathbb H}_{μλ}$ of a system of two identical bosons interacting on site and nearest-neighbor sites in the two dimensional lattice $\mathbb{Z}^2$ with interaction magnitudes $μ\in\mathbb{R}$ and $λ\in\mathbb{R},$ respectively. We completely describe the spectrum of $\hat H_{μλ}(0)$ and establish the optimal lower bound for the number of eigenvalues of $\hat H_{μλ}(K)$ outside its essential spectrum for all values of $K\in\mathbb{T}^2.$ Namely, we partition the $(μ,λ)$-plane such that in each connected component of the partition the number of bound states of $\hat H_{μλ}(K)$ below or above its essential spectrum cannot be less than the corresponding number of bound states of $\hat H_{μλ}(0)$ below or above its essential spectrum.

math-ph↗

arXiv2020

Bound states of discrete Schrödinger operators on one and two dimensional lattices

We study the spectral properties of discrete Schrödinger operator $$ \widehat H_μ=\widehat H_0 + μ\widehat{V},\qquad μ\ge0, $$ associated to a one-particle system in $d$-dimensional lattice $\mathbb{Z}^d, $ $d=1,2,$ where the non-perturbed operator $\hat H_0$ is a self-adjoint Laurent-Toeplitz-type operator generated by $\hat e:\mathbb{Z}^d\to\mathbb{C}$ and the potential $\hat V$ is the multiplication operator by $\hat v:\mathbb{Z}^d\to\mathbb{R}.$ Under certain regularity assumption on $\hat e$ and a decay assumption on $\hat v$, we establish the existence or non-existence and also the finiteness of eigenvalues of $\hat H_μ.$ Moreover, in the case of existence we study the asymptotics of eigenvalues of $\hat H_μ$ as $μ\searrow 0.$

math-ph↗

arXiv2020

Convergent expansions of eigenvalues of the generalized Friedrichs model with a rank-one perturbation

We study the existence of eigenvalues of the generalized Friedrichs model $H_μ(p)$, with a rank-one perturbation, depending on parameters $μ>0$ and $p\in\mathbb{T}^2$, and found an absolutely convergent expansions for eigenvalues at $μ(p)$, the coupling constant threshold. The expansions are highly dependent on that, whether the threshold $m(p)$ of the essential spectrum is: $(i)$ neither an threshold eigenvalue nor a threshold resonance; $(ii)$ a threshold resonance; $(iii)$ an threshold eigenvalue.

math.SP↗

arXiv2017

Bounds on the Pure Point Spectrum of Lattice Schrödinger Operators

In dimension $d\geq 3$, a variational principle for the size of the pure point spectrum of (discrete) Schrödinger operators $H(\mathfrak{e},V)$ on the hypercubic lattice $\mathbb{Z}^{d}$, with dispersion relation $\mathfrak{e}$ and potential $V$, is established. The dispersion relation $\mathfrak{e}$ is assumed to be a Morse function and the potential $V(x)$ to decay faster than $|x|^{-2(d+3)}$, but not necessarily to be of definite sign. Our estimate on the size of the pure-point spectrum yields the absence of embedded and threshold eigenvalues of $H(\mathfrak{e},V)$ for a class ot potentials of this kind. The proof of the variational principle is based on a limiting absorption principle combined with a positive commutator (Mourre) estimate, and a Virial theorem. A further observation of crucial importance for our argument is that, for any selfadjoint operator $B$ and positive number $λ>0$, the number of negative eigenvalues of $λB$ is independent of $λ$.

math-ph↗

arXiv2017

Bounds on the Discrete Spectrum of Lattice Schrödinger Operators

We discuss the validity of the Weyl asymptotics -- in the sense of two-sided bounds -- for the size of the discrete spectrum of (discrete) Schrödinger operators on the $d$--dimensional, $d\geq 1$, cubic lattice $\mathbb{Z}^{d}$ at large couplings. We show that the Weyl asymptotics can be violated in any spatial dimension $d\geq 1$ -- even if the semi-classical number of bound states is finite. Furthermore, we prove for all dimensions $d\geq 1$ that, for potentials well-behaved at infinity and fulfilling suitable decay conditions, the Weyl asymptotics always hold. These decay conditions are mild in the case $d\geq 3$, while stronger for $d=1,2$. It is well-known that the semi-classical number of bound states is -- up to a constant -- always an upper bound on the size of the discrete spectrum of Schrödinger operators if $d\geq 3$. We show here how to construct general upper bounds on the number of bound states of Schrödinger operators on $\mathbb{Z}^{d}$ from semi-classical quantities in all space dimensions $d\geq 1$ and independently of the positivity-improving property of the free Hamiltonian.

math-ph↗