= Decaying two-sided Green function of a constant-coefficient ordinary differential equation
Let $p(D)$ be a constant-coefficient differential operator, $D=d/dt$, with distinct characteristic roots $\lambda_j$ having nonzero real parts. The solution of $p(D)K=\delta$ decaying at both temporal infinities is
$$
K(t)=H(t)\sum_{\operatorname{Re}\lambda_j<0}\frac{e^{\lambda_jt}}{p'(\lambda_j)}-H(-t)\sum_{\operatorname{Re}\lambda_j>0}\frac{e^{\lambda_jt}}{p'(\lambda_j)},
$$
where $H$ is the <Heaviside step function>. This follows by <partial fraction decomposition> of $1/p(z)$ and the first-order kernels $e^{\lambda t}H(t)$ for $\operatorname{Re}\lambda<0$, or $-e^{\lambda t}H(-t)$ for $\operatorname{Re}\lambda>0$. Both first-order kernels jump by one and solve $(D-\lambda)K_\lambda=\delta$ as <distributions>.
For example, $p(D)=(D+1)(D-2)(D+2)$ gives
$$
K(t)=\begin{cases}-e^{2t}/12,&t<0,\\-e^{-t}/3+e^{-2t}/4,&t>0.\end{cases}
$$
It is continuous together with its first derivative; its second derivative jumps by one, as required by the <jump conditions for a third-order Green function>. A solution of $p(D)y=f$ is the <convolution> $K*f$ when it exists. If positive-real-part roots occur, decay at both ends generally requires a nonzero negative-time branch: this is the distinction captured by <causality versus stability of a Green function>. A causal kernel would retain growing positive-time terms and need not have a real-axis <Fourier transform>.
Back to article page
