 Causal Green function of a damped oscillator

For $L=D_t^2+2k{D}_t+k^2+{\omega }^2$, the causal Green function satisfies $LG=\delta (t)$. It is continuous at zero, with a unit jump in its first derivative. Its smooth positive-time kernel $g$ obeys $g(0)=0,g^{\prime }(0)=1$. The zero-frequency limit is $H(t)t{e}^{-kt}$. Shifting $G$ gives the zero-initial-data response to a delayed Dirac delta function.