= Solution
Impermeability requires $\widehat w(-1)=\widehat w(1)=0$. Let $c=\omega/\alpha$ with real $\alpha\ne0$, and set $a=|\alpha|$. If $c$ is nonreal, or real outside the channel, the <Rayleigh equation for inviscid shear flow> gives $\widehat w''-a^2\widehat w=0$ everywhere. The two wall conditions then force $\widehat w=0$. In fact there is no nonzero globally smooth eigenfunction for any $c$: the complete family is the <inviscid Couette continuous spectrum>.
For each $\xi\in(-1,1)$, permit a localized <vorticity sheet>. Let $G_a$ be the <Dirichlet Green function> satisfying $(D^2-a^2)G_a(z,\xi)=\delta(z-\xi)$. An explicit generalized <eigenfunction> is
$$
\boxed{\widehat w_\xi(z)=G_a(z,\xi)
=-\frac{\sinh[a(z_<+1)]\sinh[a(1-z_>)]}{a\sinh(2a)},\qquad
\omega_\xi=\alpha\xi,}
$$
where $z_<=\min(z,\xi)$ and $z_>=\max(z,\xi)$. It vanishes at both walls, is continuous at $z=\xi$, and has derivative jump $[\partial_zG_a]_{\xi^-}^{\xi^+}=1$. Consequently
$$
(z-\xi)(D^2-a^2)\widehat w_\xi=(z-\xi)\delta(z-\xi)=0,
$$
using the <Dirac delta multiplication identity>. These are <vorticity-sheet eigenfunctions of inviscid Couette flow>, understood as <generalized eigenfunctions>, not as a discrete smooth <Sturm-Liouville eigenfunction expansion>.
To see completeness, define the vorticity variable $q=(D^2-a^2)w$. Its evolution is $(\partial_t+i\alpha z)q=0$, so for any admissible initial vorticity $q_0$,
$$
\boxed{w(z,t)=\int_{-1}^1G_a(z,\xi)q_0(\xi)e^{-i\alpha\xi t}\,d\xi.}
$$
The homogeneous Dirichlet problem for $D^2-a^2$ has only the zero solution, so this inversion reconstructs every initial vertical-velocity field in its usual function space. The generalized frequencies fill the interval with endpoints $\pm\alpha$; the endpoint values are understood as the closure of the <continuous spectrum>.
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