Essential and Point Spectrum of a Model Hamiltonian Associated with the Three-particle Quantum Systems on a Lattice

In this paper, we investigate the tensor sum $$H_{\mu,\lambda}$$ with $$\mu,\lambda>0$$ , constructed from two Friedrichs models perturbed by rank two operators. The Hamiltonian $$H_{\mu,\lambda}$$ describes a system of three quantum particles on a d-dimensional lattice. We define a set of parameters $$\mu$$ and $$\lambda$$ such that the essential spectrum of the Hamiltonian $$H_{\mu,\lambda}$$ is given by the union of one, two, or three closed intervals. The point spectrum of the Hamiltonian $$H_{\mu,\lambda}$$ is investigated through finite integrals appearing in the Fredholm determinant associated with the Friedrichs model.

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