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

• Published in: arXiv.org 2021-01
• Main Authors: Lakaev, Saidakhmat, Kholmatov, Shokhrukh, Khamidov, Shakhobiddin
• Format: Article
• Language: English
• Subjects: Bosons; Eigenvalues; Lower bounds; Partitions
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2101.05109

We study the discrete spectrum of the two-particle Schr\"odinger operator \(\hat H_{\mu\lambda}(K),\) \(K\in\mathbb{T}^2,\) 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 \(\hat H_{\mu\lambda}(0)\) and establish the optimal lower bound for the number of eigenvalues of \(\hat H_{\mu\lambda}(K)\) outside its essential spectrum for all values of \(K\in\mathbb{T}^2.\) Namely, we partition the \((\mu,\lambda)\)-plane such that in each connected component of the partition the number of bound states of \(\hat H_{\mu\lambda}(K)\) below or above its essential spectrum cannot be less than the corresponding number of bound states of \(\hat H_{\mu\lambda}(0)\) below or above its essential spectrum.