The goal of this talk is to discuss spectral theory for quantum Hamiltonians describing  a system of two  identical spinless bosons on the two-dimensional lattice particles.

We assume that the particles are interact via on-site and first and second nearest-neighboring site interactions between the bosons are only nontrivial and that these interactions are of magnitudes \(\gamma\), \(\lambda\), and \(\mu\), respectively.

Introducing the quasi-momentum \(k\in (-\pi,\pi]^2\)  for the system of two identical bosons,  and decomposing the  Hamiltonians into the direct von Neumann integral reduces the problem to the study  two-particle fiber lattice Schrödinger operators that depend on the two-particle quasi-momentum \(k\).

In this model, we determine both the exact number and location of the eigenvalues for the lattice Schrödinger operator \(H_{\gamma\lambda\mu}(0)\), depending on the interaction parameter \(\gamma\), \(\lambda\) and \(\mu\).

We find a partition of the \((\gamma,\lambda,\mu)\)-space into connected components such that, in each connected component, the two-boson Schrödinger operator corresponding to the zero quasi-momentum of the center of mass has a definite (fixed) number of eigenvalues, which are situated below the bottom of the essential (continuous) spectrum and above its top. Moreover,  for each connected component, a sharp lower bound is established on the number of isolated eigenvalues for the two-boson Schrödinger operator corresponding to any admissible nonzero value of the center-of-mass quasimomentum.

We will reveal the mechanisms of emergence and absorption of the eigenvalues at the thresholds of the essential (continuous) spectrum of \(H_{\gamma\lambda\mu}(0)\), as the parameters \(\gamma\), \(\lambda\) and \(\mu\) vary.

In addition, we have identified the sufficient conditions for the exact number of eigenvalues of the lattice Schrödinger operator for all values of the interaction parameters.