Causal Green function

A causal Green function vanishes before the source time and has the derivative jump required by a delta source.

With inverse Fourier transform $G(t)=\int G(\omega )e^{-i\omega t}d\omega /(2\pi )$, a tempered retarded response vanishes for $t<0$ and is analytic in the upper frequency half-plane with suitable large-frequency growth bounds. Poles strictly below the real axis describe decaying modes; upper-half-plane poles obstruct a bounded retarded real-axis Fourier response. However causality alone does not imply stability: a retarded exponentially growing Green function can exist without an ordinary Fourier transform. Its Laplace transform inversion contour must pass above the growth poles in the frequency plane. Moving the contour to the real axis changes the boundary condition and can introduce a pre-response.

For $\mathcal{L}=D^3+a_2{D}^2+a_{1}D+a_0$ with smooth coefficients and leading coefficient one, a Green function solving $\mathcal{L}G={\delta }_{\xi }$ is continuous together with its first derivative at an interior source, and its second derivative jumps by one. Distributional differentiation shows why: a jump in $G$ would create a delta second derivative, a jump in $G^{\prime }$ a delta first derivative, and a jump in $G^{\prime \prime }$ the required delta. For a causal initial-value Green function, take the left branch zero and choose the right branch with initial values $0,0,1$ at the source.