arXiv subjects

Mukhayyo O. Akhmadova

Publications and source records attributed to Mukhayyo O. Akhmadova.

2 recordsLinked to original sources

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

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 $H_{γλμ}(0)$.

math-ph↗