Spectral analysis of two-particle Hamiltonians with short-range interactions

We analyze the spectral characteristics of lattice Schrödinger operators, denoted as $H_{\gamma\lambda\mu}(K)$, $K\in(-\pi,\pi]^3$, which represent a system of two identical bosons existing on $\mathbb{Z}^3$ lattice. The model includes onsite and nearest-neighbor interactions, parameterized by $\gamma,\lambda,\mu\in\mathbb{R}$. Our study of $H_{\gamma\lambda\mu}(0)$ reveals an invariant subspace on which its restricted form, $H_{\lambda\mu}^{\mathrm{ea}}(0)$, is solely dependent on $\lambda$ and $\mu$. To elucidate the mechanisms of eigenvalue birth and annihilation for $H_{\lambda\mu}^{\mathrm{ea}}(0)$, we define a critical operator. A detailed criterion is subsequently developed within the plane spanned by $\lambda$ and $\mu$. This involves: (i) the derivation of smooth critical curves that mark the onset of criticality for the operator, and (ii) the proof of exact conditions for the existence of precisely $\alpha$ eigenvalues below and $\beta$ eigenvalues above the essential spectrum, where $\alpha,\beta\in\{0,1,2\}$ and $\alpha+\beta\le2$.