Bose–Hubbard models with on-site and nearest-neighbor interactions: exactly solvable case,Journal of Physics A: Mathematical and Theoretical

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Bose–Hubbard models with on-site and nearest-neighbor interactions: exactly solvable case
Journal of Physics A: Mathematical and Theoretical ( IF 2.0 ) Pub Date : 2021-05-21 , DOI: 10.1088/1751-8121/abfcf4
Saidakhmat N Lakaev ₁ , Shokhrukh Yu Kholmatov ₂ , Shakhobiddin I Khamidov ₁

Affiliation

We study the discrete spectrum of the two-particle Schrdinger operator ${\hat{H}}_{\mu \lambda }\left(K\right)$ , $K\in {\mathbb{T}}^{2}$ , associated to the Bose–Hubbard Hamiltonian ${\hat{\mathbb{H}}}_{\mu \lambda }$ 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 $\mu \in \mathbb{R}$ and $\lambda \in \mathbb{R}$ , respectively. We completely describe the spectrum of ${\hat{H}}_{\mu \lambda }\left(0\right)$ and establish the optimal lower bound for the number of eigenvalues of ${\hat{H}}_{\mu \lambda }\left(K\right)$ 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 ${\hat{H}}_{\mu \lambda }\left(K\right)$ below or above its essential spectrum cannot be less than the corresponding number of bound states of ${\hat{H}}_{\mu \lambda }\left(0\right)$ below or above its essential spectrum.

中文翻译：

具有现场和最近邻相互作用的 Bose-Hubbard 模型：完全可解的情况

我们研究了两粒子薛定谔算子的离散频谱 ${\hat{H}}_{\mu \lambda }\left(K\right)$ ， $K\in {\mathbb{T}}^{2}$ 相关联的玻色-哈巴德哈密顿 ${\hat{\mathbb{H}}}_{\mu \lambda }$ 两个相同的玻色子在二维网格上的网站，最近邻部位相互作用的系统 ${\mathbb{Z}}^{2}$ 与互动幅度 $\mu \in \mathbb{R}$ 和 $\lambda \in \mathbb{R}$ 分别。我们完整地描述了的频谱， ${\hat{H}}_{\mu \lambda }\left(0\right)$ 并 ${\hat{H}}_{\mu \lambda }\left(K\right)$ 为的所有值的基本频谱之外的特征值数量建立了最优下界 $K\in {\mathbb{T}}^{2}$ 。也就是说，我们对 ( μ , λ ) 平面进行分区，使得在分区的每个连通分量中， ${\hat{H}}_{\mu \lambda }\left(K\right)$ 低于或高于其本征谱不能少于低于或高于其本征谱的相应束缚态数 ${\hat{H}}_{\mu \lambda }\left(0\right)$ 。

更新日期：2021-05-21

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