Solving for Multiple Bound States Contained Within the Continuum of the Friedrichs Model

The Friedrichs model consists of an interaction between the discrete states of a quantum system (say, a set of atoms) and a set of continuous states. The interaction produces a resonance, resulting in complex eigenenergies. The resonance appears as a pole of the analytic continuation of the reduced resolvent corresponding to the interacting Hamiltonian. As the imaginary component approaches 0, the eigenstate's lifetime becomes infinite, forming a bound state in the continuum. We solve for a set of parameters that generates multiple resonances with infinite lifetime within the model.