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For inviscid plane Couette flow $U=z$ between impermeable walls at $z=\pm 1$, the nonzero real streamwise wavenumber $\alpha$ gives no nontrivial smooth discrete vertical-velocity modes. Instead, the vorticity variable $q=(D^2-{\alpha }^2)w$ obeys $({\partial }_t+i\alpha z)q=0$. Localized vorticity sheets give generalized eigenfunctions with real frequencies ${\omega }_{\xi }=\alpha \xi$. Their closure forms a neutral continuous spectrum.

A Jordan chain of a linear operator $T$ for an eigenvalue $\lambda$ is a list of nonzero vectors satisfying $(T-\lambda I)v_1=0$ and $(T-\lambda I)v_j=v_{j-1}$ for $2\le j\le r$. These vectors are linearly independent: in a vanishing linear combination, applying the largest relevant power of $T-\lambda I$ isolates the last coefficient times $v_1$, then descending induction removes all coefficients. On their span the operator has one Jordan block. This is different from a generalized eigenfunction that solves $(T-\lambda I)u=0$ outside the original function space.

For $E>0$, put $k=\sqrt{2mE}$ and $q=k/\hslash >0$. Outside the support of the potential energy, a generalized eigenfunction of the Hamiltonian operator has the plane wave formHere $R_0$ denotes the support radius, to distinguish it from the quantum reflection probability $R$. The momenta are $\pm k$ and the velocities are $\pm k/m$. Thus $A,D$ describe incoming scattering channels from the left and right, and $B,C$ describe outgoing scattering channels towards the left and right. These energy eigenfunctions are not square-integrable functions; normalizable incoming wave packets are formed by superposition over energy, and their passage through the interaction region gives the physical quantum scattering experiment.

For incidence from the left, normalize $A=1,D=0$ and write $B=r_L,C=t_L$. The quantum reflection amplitude is $r_L$ and the transmission amplitude is $t_L$. The ratios of outgoing to incident probability currents defineNo extra velocity ratio occurs because both exterior regions have the same potential energy and mass. The scattering matrix, with a fixed choice of channel order, isIts first and second columns are the responses to unit incidence from the left and right, respectively. For a reflection-symmetric potential energy, these conventions give $r_L=r_R=r$ and $t_L=t_R=t$.

Impermeability requires $\hat{w}(-1)=\hat{w}(1)=0$. Let $c=\omega /\alpha$ with real $\alpha \ne 0$, and set $a=|\alpha |$. If $c$ is nonreal, or real outside the channel, the Rayleigh equation for inviscid shear flow gives ${\hat{w}}^{\prime \prime }-a^2\hat{w}=0$ everywhere. The two wall conditions then force $\hat{w}=0$. 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 iswhere $z_{<}=\min (z,\xi )$ and $z_{>}=\max (z,\xi )$. It vanishes at both walls, is continuous at $z=\xi$, and has derivative jump $[{\partial }_z{G}_a{]}_{{\xi }^{-}}^{{\xi }^{+}}=1$. Consequentlyusing 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 $q_0$,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.

The wavefunction is a plane wave and a simultaneous formal eigenfunction of the momentum operator and the time-translation energy operator:Its momentum and energy are therefore definite, while its probability density is spatially uniform:For a particle of mass $m$, the probability current is $j=(\hslash /m)\mathrm{Im}(\overline{\varphi }{\partial }_x\varphi )=\hslash k{A}^2/m$, directed according to the sign of $k$.

On the whole real line, a nonzero constant-amplitude plane wave is not square-integrable and cannot be normalised to total probability one. It represents an idealised momentum eigenstate, interpreted through generalized eigenfunctions or as a limit of wave packets. Also, arbitrary $k,E$ do not automatically solve the Schrödinger equation for a specified potential energy: for a free particle, $E={\hslash }^2{k}^2/(2m)$, or this value plus the constant potential energy for a constant-potential region. If $A=0$, the zero function is not a physical quantum state.

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.