Bound-state eigenenergy outside and inside the continuum for unstable multilevel systems

The eigenvalue problem for the dressed bound state of unstable multilevel systems is examined both outside and inside the continuum, based on the $N$-level Friedrichs model, which describes the couplings between the discrete levels and the continuous spectrum. It is shown that a bound-state eigenenergy always exists below each of the discrete levels that lie outside the continuum. Furthermore, by strengthening the couplings gradually, the eigenenergy corresponding to each of the discrete levels inside the continuum finally emerges. On the other hand, the absence of the eigenenergy inside the continuum is proved in weak but finite coupling regimes, provided that each of the form factors that determine the transition between some definite level and the continuum does not vanish at that energy level. An application to the spontaneous emission process for the hydrogen atom interacting with the electromagnetic field is demonstrated.

Figure 1

${\omega }_3-{\kappa }_n\left(0\right)$ for $n=1,2,3$ (three solid lines) for a three-level system with form factors (32), plotted against $\lambda$, and ${\omega }_3-{\omega }_1$, ${\omega }_3-{\omega }_2$ (two dashed lines), and ${\omega }_3$ (dot-dashed line), for reference, where ${\omega }_3-{\omega }_1>{\omega }_3>{\omega }_3-{\omega }_2$. Three different regions are distinguished, corresponding to the number of solid lines satisfying ${\omega }_3-{\kappa }_n\left(0\right)>{\omega }_3$, that is, just the number of negative eigenenergies of $H$, by Theorem III.4.

Figure 2

${\omega }_3-{\kappa }_n\left(E\right)$ for $n=1,2,3$ (three solid lines) for a three-level system of the form factors (32), ${\omega }_3-E$ (short-dashed line), and ${\omega }_3-{\omega }_1$ and ${\omega }_3-{\omega }_2$ (two dashed lines). We plot them in $\lambda =0.1$, 0.7, and $10.0$, in (a), (b), and (c), respectively, choosing the parameters as ${\omega }_1∕\Lambda =-0.01$, ${\omega }_2∕\Lambda =0.01$, and ${\omega }_3∕\Lambda =0.02$. (a) For a relatively small $\lambda$, only ${\omega }_3-{\kappa }_1\left(E\right)$ intersects ${\omega }_3-E$ at $E∕\Lambda \simeq -0.02$, which is predicted by Theorem III.4. Thus there is one negative eigenenergy. One also sees that ${\omega }_3-{\kappa }_1\left(E\right)$ and ${\omega }_3-{\kappa }_2\left(E\right)$ still lie closely above the asymptotes ${\omega }_3-{\omega }_1$ and ${\omega }_3-{\omega }_2$, respectively. (b) Both ${\omega }_3-{\kappa }_1\left(E\right)$ and ${\omega }_3-{\kappa }_2\left(E\right)$ intersect ${\omega }_3-E$ in the vicinity of $E=-0.3$ and 0.0, respectively, so that there are two negative eigenenergies. (c) All ${\omega }_3-{\kappa }_n\left(E\right)$ for $n=1$, 2, and 3 intersect ${\omega }_3-E$, and thus three negative eigenenergies exist. In this figure, only two intersections for $n=2$ and 3 are depicted.