On the generalized Friedrichs-Lee model with multiple discrete and continuous states

TL;DR

This work extends the Friedrichs-Lee model to handle multiple discrete and continuum channels while preserving exact solvability. It introduces separable continuum-continuum interactions and a finite-rank expansion approach to approximate general potentials, enabling explicit construction of the $S$-matrix and the study of pole structures across multiple Riemann sheets. The authors derive matrix formulations for the continuum and discrete sectors, show coupled-channel unitarity is automatically satisfied, and demonstrate methods to obtain continuum and discrete eigenstates, as well as approximate solutions when non-separable potentials are present. The framework is applicable across hadron and nuclear physics, optics, and cold-atom systems, offering a controllable path to analyze resonance phenomena with multiple interfering states. The work also discusses pole trajectories and the qualitative impact of various couplings on bound, virtual, and resonant states, providing practical guidance for interpreting spectra in coupled-channel scenarios.

Abstract

In this study, we present several improvements of the non-relativistic Friedrichs-Lee model with multiple discrete and continuous states and still retain its solvability. Our findings establish a solid theoretical basis for the exploration of resonance phenomena in scenarios involving multiple interfering states across various channels. The scattering amplitudes associated with the continuum states naturally adhere to coupled-channel unitarity, rendering this framework particularly valuable for investigating hadronic resonant states appearing in multiple coupled channels. Moreover, this generalized framework exhibits a wide-range applicability, enabling investigations into resonance phenomena across diverse physical domains, including hadron physics, nuclear physics, optics, and cold atom physics, among others.

Figures (3)

• Figure 1: The integral contour for the resonance solution.
• Figure 2: A monotonic increasing function $\zeta(E)$ which satisfies $\zeta(\mu)=0$. The solution for $\zeta(E)+\chi(E)=0$ is smaller than $\mu$ if $\chi(E)$ is a monotonic increasing positive function.
• Figure 3: The general behavior of the real part of $G(E)=\int_{a}^\infty d\omega \frac{|f(\omega)|^2}{E-\omega+i\epsilon}$ function, using $|f(\omega)|^2=\sqrt{\omega-a} e^{-\omega/\Lambda}$, with $\Lambda=5$, $a=0$. If the interested energy region is much smaller than $\Lambda$, then $\mathrm {Re} G(E)<0$.