Threshold Resonances, Critical Couplings, and Eigenvalue Bounds for Two-Particle Operators on $\mathbb{Z}^3$

We study a family of lattice Schr\"odinger operators $H_{\mu_1\mu_2}(K)$ describing two identical bosons on the three-dimensional cubic lattice $\mathbb{Z}^3$, where $K \in \mathbb{T}^3$ is the quasi-momentum, and $\mu_1, \mu_2 \in \mathbb{R}$ 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 $H_{\mu_1\mu_2}(0)$. A detailed spectral analysis of the restriction of $H_{\mu_1\mu_2}(0)$ to one of these subspaces reveals two smooth critical curves in the $(\mu_1, \mu_2)$-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 $\mu_2$-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, $H_{\mu_1\mu_2}(0)$ may possess up to three bound states located either below or above the essential spectrum, with numbers $(\alpha,\beta)$ satisfying $\alpha + \beta < 3$ or $(\alpha,\beta) \in \{(3,0),(0,3)\}$. 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 $H_{\mu_1\mu_2}(K)$.