Let me expand a little more on what Craig Thone just said :
Considere the energy/frequency-dependent Green function : $$ \tilde{G}(\omega)=\frac{1}{\omega-(a-\mathrm{i}b)} $$ with one single pole in $\omega=a-\mathrm{i}b$ (with $b>0$), which is Fourier transform of the time-dependent $G(t)$ Green function such as : $$ G(t)=\int\frac{\mathrm{d}\omega}{2\pi}\frac{e^{\mathrm{i}\omega t}}{\omega-(a-\mathrm{i}b)} $$ One can show, using complexe analysis (I can eventually show some details about that if needed), that it computes to : $$ G(t)=\mathrm{i}\,e^{\mathrm{i}at-bt}\Theta(t) $$ where $\Theta$ is an Heaviside step-function.
It means that :
- The real part $a$ of the pole gives an oscillatory behavior to the solution. In the context of quantum systems, $a$ often refers to as an eigen-energy. 2. The role the imaginary part $b$ in twofold : - It gives a damped behavior to the solution. In the context of quantum systems, $b$ will describe how one state, which is not an eigen-state of the system, will be depleted as a function of time in a superposition of eigen-states. In the limit of a perturbation theory, $b$ is the same rate given by the Fermi Golden Rule. - The fact that $b>0$ ensure that the function $\tilde{G}(\omega)$ has no pole in the upper complexe plane (i.e. $\forall\omega\in\mathbb{C}, \Im({\omega})>0$). This analicity of $\tilde{G}$ implies by the Cauchy's integral theorem that : $$ \forall t<0,\,G(t)=0 $$ which is needed to ensure the causality of the dynamics. To ensure such property, an Heaviside step-function $\Theta$ can be added to $G(t)$.