 Jump conditions for a third-order Green function

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.