Vorticity-sheet eigenfunction of inviscid Couette flow (${\hat{w}}_{\xi }=G_{|\alpha |}(z,\xi )$)

Let $G_a$ be the Dirichlet Green function of $D^2-a^2$ on $[-1,1]$, with $a=|\alpha |$:It is continuous and has derivative jump one, so $(D^2-a^2)G_a=\delta (z-\xi )$. The Dirac delta multiplication identity gives $(z-\xi )\delta (z-\xi )=0$, proving it is a generalized eigenfunction of the inviscid Couette continuous spectrum. Integrating $G_a(z,\xi )q_0(\xi )e^{-i\alpha \xi t}$ reconstructs the evolving vertical velocity from its initial vorticity.