Perturbation theory on generalized quantum mechanical systems

We present a general formalism for doing the perturbation theory in the complex energy plane, where the notion of the generalized quantum mechanical systems is used. This formalism is applied to the Friedrichs-Lee model. It reproduces the results of the exact solution, where the spectrum of the generalized quantum mechanical system consists of a discrete complex energy pole and a continuum spectrum (which passes below this discrete pole) in the complex energy plane. We also investigate the role of the “complex delta” function in the description of a resonance state. The unboundedness of the spectrum appears to be the very ingredient needed to give rise to a pure exponential decay.