The Friedrichs Model and its use in resonance phenomena

TL;DR

This paper explores various Friedrichs models to understand resonance phenomena, analyzing their mathematical properties and applications in quantum field theory, including models with multiple levels, degeneracies, and continuous spectra.

Contribution

It systematically relates different Friedrichs models and demonstrates their use in describing resonance phenomena in quantum systems, including new generalizations and mathematical foundations.

Findings

Resonance in simple and degenerate Friedrichs models analyzed.

Oscillatory behavior of probability amplitudes in N-level models shown.

Generalizations applicable to quantum field theory introduced.

Abstract

We present here a relation of different types of Friedrichs models and their use in the description and comprehension of resonance phenomena. We first discuss the basic Friedrichs model and obtain its resonance in the case that this is simple or doubly degenerated. Next, we discuss the model with $N$ levels and show how the probability amplitude has an oscillatory behavior. Two generalizations of the Friedrichs model are suitable to introduce resonance behavior in quantum field theory. We also discuss a discrete version of the Friedrichs model and also a resonant interaction between two systems both with continuous spectrum. In an Appendix, we review the mathematics of rigged Hilbert spaces.