Unified description of resonance and decay phenomena

TL;DR

This paper presents a unified formalism using non-Hermitian Hamiltonians to describe resonance and decay phenomena, deriving decay rates and relating lifetime to resonance width across different regimes.

Contribution

It introduces a comprehensive approach that applies to isolated and overlapping resonances, deriving a fundamental relation between lifetime and width, and connecting time asymmetry with the time operator.

Findings

Derived an expression for decay rates of resonance states.

Established the fundamental relation τ = ħ / Γ for isolated resonances.

Showed the decay rate decreases monotonously in overlapping regimes.

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.