Threshold Resonances, Critical Couplings, and Eigenvalue Bounds for Two-Particle Operators on $\mathbb{Z}^3$
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2025
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| author | Lakaev, Saidakhmat N. Abduvayitov, Saidakbar S. Lakaev, Shuhrat S. |
| author_facet | Lakaev, Saidakhmat N. Abduvayitov, Saidakbar S. Lakaev, Shuhrat S. |
| contents | We study a family of lattice Schrödinger operators $H_{μ_1μ_2}(K)$ describing two identical bosons on the three-dimensional cubic lattice $\mathbb{Z}^3$, where $K \in \mathbb{T}^3$ is the quasi-momentum, and $μ_1, μ_2 \in \mathbb{R}$ are coupling constants corresponding to on-site and nearest-neighbour interactions, respectively.
We show that the Hilbert space $L^{2,\mathrm{e}}(\mathbb{T}^3)$ decomposes into three mutually orthogonal subspaces, each invariant under $H_{μ_1μ_2}(0)$. A detailed spectral analysis of the restriction of $H_{μ_1μ_2}(0)$ to one of these subspaces reveals two smooth critical curves in the $(μ_1, μ_2)$-plane, separating regions where the number of eigenvalues below the essential spectrum remains constant. For the restrictions to the other two subspaces, we identify a critical point on the $μ_2$-axis that partitions it into intervals with a constant number of eigenvalues below the essential spectrum.
Analogously, two additional critical curves and one critical point determine regions and intervals where the number of eigenvalues above the essential spectrum is constant. In particular, for suitable parameter values, $H_{μ_1μ_2}(0)$ may possess up to three bound states located either below or above the essential spectrum, with numbers $(α,β)$ satisfying $α+ β< 3$ or $(α,β) \in \{(3,0),(0,3)\}$.
Here, by \emph{eigenvalue bounds} we mean both the possible locations of eigenvalues outside the essential spectrum and the maximum number of such eigenvalues for given coupling parameters. Finally, we extend the analysis to arbitrary quasi-momentum $K \in \mathbb{T}^3$, obtaining general lower bounds for the number of eigenvalues of $H_{μ_1μ_2}(K)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_14321 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Threshold Resonances, Critical Couplings, and Eigenvalue Bounds for Two-Particle Operators on $\mathbb{Z}^3$ Lakaev, Saidakhmat N. Abduvayitov, Saidakbar S. Lakaev, Shuhrat S. Spectral Theory Primary: Spectral Theory (math.SP) Secondary: 81Q10, 81Q35, 47A10, 47B37 F.2.2 We study a family of lattice Schrödinger operators $H_{μ_1μ_2}(K)$ describing two identical bosons on the three-dimensional cubic lattice $\mathbb{Z}^3$, where $K \in \mathbb{T}^3$ is the quasi-momentum, and $μ_1, μ_2 \in \mathbb{R}$ are coupling constants corresponding to on-site and nearest-neighbour interactions, respectively. We show that the Hilbert space $L^{2,\mathrm{e}}(\mathbb{T}^3)$ decomposes into three mutually orthogonal subspaces, each invariant under $H_{μ_1μ_2}(0)$. A detailed spectral analysis of the restriction of $H_{μ_1μ_2}(0)$ to one of these subspaces reveals two smooth critical curves in the $(μ_1, μ_2)$-plane, separating regions where the number of eigenvalues below the essential spectrum remains constant. For the restrictions to the other two subspaces, we identify a critical point on the $μ_2$-axis that partitions it into intervals with a constant number of eigenvalues below the essential spectrum. Analogously, two additional critical curves and one critical point determine regions and intervals where the number of eigenvalues above the essential spectrum is constant. In particular, for suitable parameter values, $H_{μ_1μ_2}(0)$ may possess up to three bound states located either below or above the essential spectrum, with numbers $(α,β)$ satisfying $α+ β< 3$ or $(α,β) \in \{(3,0),(0,3)\}$. Here, by \emph{eigenvalue bounds} we mean both the possible locations of eigenvalues outside the essential spectrum and the maximum number of such eigenvalues for given coupling parameters. Finally, we extend the analysis to arbitrary quasi-momentum $K \in \mathbb{T}^3$, obtaining general lower bounds for the number of eigenvalues of $H_{μ_1μ_2}(K)$. |
| title | Threshold Resonances, Critical Couplings, and Eigenvalue Bounds for Two-Particle Operators on $\mathbb{Z}^3$ |
| topic | Spectral Theory Primary: Spectral Theory (math.SP) Secondary: 81Q10, 81Q35, 47A10, 47B37 F.2.2 |
| url | https://arxiv.org/abs/2511.14321 |