On the existence of the maximum number of isolated eigenvalues for a lattice Schrödinger operator

This paper presents a detailed spectral analysis of the discrete Schrodinger operator¨ H γλ µ ( K ), which describes a system of two identical bosons on a two-dimensional lattice, Z 2 . The operator’s family is parameterized by the quasi-momentum K ∈ T 2 and real interaction strengths: γ for on-site, λ for nearest-neighbor, and µ for next-nearest-neighbor interactions. A key finding of our study is that, under specific conditions on the interaction parameters, the operator H γλ µ ( 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 ∈ T 2 .