Abstract:
This paper presents a detailed spectral analysis of the discrete Schrödinger operator $H_{\gamma\lambda\mu}(K)$, which describes a system of two identical bosons on a two-dimensional lattice, $\mathbb{Z}^2$. The operator’s family is parameterized by the quasi-momentum $K \in \mathbb{T}^2$ and real interaction strengths: $\gamma$ for on-site, $\lambda$ for nearestneighbor, and $\mu$ for next-nearest-neighbor interactions. A key finding of our study is that, under specific conditions on the interaction parameters, the operator $H_{\gamma\lambda\mu}(K)$ consistently possesses a total of seven eigenvalues that lie either below the bottom or above the top of its essential spectrum, over all $K \in \mathbb{T}^2$.
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