Shakhobiddin Khamidov

arXiv2024

The number and location of two particle Schrödinger operators on a lattice

We study the Schrödinger operators ${H}_{λμ}(K)$ with $K\in\mathbb{T}^2$ being the fixed quasimomentum of a pair of particles, associated with a system of two arbitrary particles on a two-dimensional lattice $\mathbb{Z}^2$ with on-site and nearest-neighbor interactions of strengths $λ\in\mathbb{R}$ and $μ\in\mathbb{R}$, respectively. We divide the $(λ,μ)$-plane of parameters $λ$ and $μ$ into connected components, such that in each component, the Schrödinger operator $H_{λμ}(0)$ has a fixed number of eigenvalues. These eigenvalues are located both below the bottom of the essential spectrum and above its top. Additionally, we establish a sharp lower bound for the number of isolated eigenvalues of $H_{λμ}(K)$ within each connected component.

math-ph↗

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↗