Characterization of the Point Spectrum of a Three-Particle Lattice Model Hamiltonian

Abstract

In the present work, we investigate a lattice Hamiltonian \(H_{\mu,\lambda}\), \(\mu,\lambda>0\), describing a three-particle quantum system with non-local interaction terms. The operator under consideration can be viewed as a tensor sum coupling of two generalized Friedrichs models perturbed by a rank-two operator and is associated with particles propagating on a \(d\)-dimensional lattice. The study is mainly focused on the spectral analysis of \(H_{\mu,\lambda}\), especially on the characterization of its point spectrum. Employing the Fredholm determinant technique together with methods from the spectral theory of operator matrices, we obtain criteria describing the existence and localization of eigenvalues of \(H_{\mu,\lambda}\). In addition, we examine how the point spectrum depends on the interaction parameters \(\mu\) and \(\lambda\). Moreover, sufficient conditions guaranteeing the existence of one or two isolated simple eigenvalues are established. The obtained results contribute to a deeper understanding of spectral phenomena for lattice quantum systems with non-local interactions and may be useful in further investigations of multi-particle models in mathematical physics.