Asymptotic expansion of Fredholm determinant associated to a family of Friedrichs models arising in quantum mechanics

In this paper, we consider a family of Friedrichs models that arise in quantum mechanics and corresponding to the Hamiltonian of a two-particle system on a one-dimensional lattice. The number, location, and existence conditions of eigenvalues of this family were analyzed. An asymptotic expansion for the associated Fredholm determinant in a neighborhood of the origin has been derived.