Number of bound states of the Hamiltonian of a lattice two-boson system with interactions up to the next neighbouring sites
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929426574344192 |
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| author | Lakaev, Saidakhmat N. Khamidov, Shakhobiddin I. Akhmadova, Mukhayyo O. |
| author_facet | Lakaev, Saidakhmat N. Khamidov, Shakhobiddin I. Akhmadova, Mukhayyo O. |
| contents | 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 $H_{γλμ}(K)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_13552 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Number of bound states of the Hamiltonian of a lattice two-boson system with interactions up to the next neighbouring sites Lakaev, Saidakhmat N. Khamidov, Shakhobiddin I. Akhmadova, Mukhayyo O. Mathematical Physics 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 $H_{γλμ}(K)$. |
| title | Number of bound states of the Hamiltonian of a lattice two-boson system with interactions up to the next neighbouring sites |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2407.13552 |