Unified description of resonance and decay phenomena

Abstract

In the Feshbach projection operator formalism, resonance as well as decay phenomena are described by means of the complex eigenvalues and eigenfunctions of the non-Hermitian Hamilton operator $H_{\rm eff}$ that appears in an intermediate stage of the formalism. The formalism can be applied for the description of isolated resonances as well as for resonances in the overlapping regime. Time asymmetry is related to the time operator which is a part of $H_{\rm eff}$. An expression for the decay rates of resonance states is derived. For isolated resonance states $\lambda$, this expression gives the fundamental relation $\tau_\lambda = \hbar / \Gamma_\lambda$ between life time and width of a resonance state. A similar relation holds for the average values obtained for narrow resonances superposed by a smooth background term. In the cross over between these two cases (regime of overlapping resonances), the decay rate decreases monotonously as a function of increasing time.