Unphysical energy sheets and resonances in the Friedrichs-Faddeev model

Abstract: We consider the Friedrichs-Faddeev model in the case where the kernel of the potential operator is holomorphic in both arguments on a certain domain of $\mathbb{C}$. For this model we, first, study the structure of the $T$- and $S$-matrices on unphysical energy sheet(s). To this end, we derive representations that explicitly express them in terms of these same operators considered exclusively on the physical sheet. Furthermore, we allow the Friedrichs-Faddeev Hamiltonian undergo a complex deformation (or even a complex scaling/rotation if the model is associated with an infinite interval). Isolated non-real eigenvalues of the deformed Hamiltonian are called the deformation resonances. For a class of perturbation potentials with analytic kernels, we prove that the deformation resonances do correspond to the scattering matrix resonances, that is, they represent the poles of the scattering matrix analytically continued to the respective unphysical sheet.