Fourth-order Green kernel with two decay scales ($K(x)=\frac{e^{-a|x|}/a-e^{-b|x|}/b}{2(b^2-a^2)}$)

For $0<a<b$, the displayed decaying Green function solves $(-{\partial }_x^2+a^2)(-{\partial }_x^2+b^2)K={\delta }_0$ on the real line. Its Fourier transform isInvert each term using the one-dimensional modified Helmholtz Green function. Then a decaying solution with forcing $h$ is the convolution $K\ast h$. For $a=1,b=2$, this gives $K=e^{-|x|}/6-e^{-2|x|}/12$ for the operator ${\partial }_x^4-5{\partial }_x^2+4$. Its first-derivative jump cancels, while its third-derivative jump is one, providing the unit Dirac delta distribution required by the fourth-order operator.