Interaction of an isolated state with an infinite quantum system containing several one-parameter eigenvalue bands: II. time-dependent case

Abstract

A new method for the exact solution of the interaction of an isolated state $$\left| \Theta \right\rangle $$ with an infinite dimensional quantum system § ∞ containing several one-parameter eigenvalue bands $$\lambda _\nu (k)\in I_\nu \equiv \left[ {a_\nu ,b_\nu } \right]$$ is developed. Unlike standard perturbation expansion approach, this method produces correct results however strong the interaction between the state $$\left| \Theta \right\rangle $$ and the system $$\S_\infty ^{\,b} $$ . It is shown that in the case of the weak interaction this method correctly reproduces standard results obtained within the formalism of the perturbation expansion method. In particular, due to the interaction with the system $$\S_\infty ^{\,b} $$ , eigenvalue E of the state $$\left| \Theta \right\rangle $$ shifts to a new position. In addition, if this eigenvalue is embedded inside the range $$D=\bigcup\nolimits_\nu {I_\nu } $$ of the unperturbed eigenvalues, this shifted eigenvalue broadens and spectral distribution of the state $$\left| \Theta \right\rangle $$ has the shape of the universal resonance curve. However, if the interaction is strong, one finds much more complex and much more complicated behavior