The Orr Sommerfeld equation is as follows $$\psi''-k^2 \psi - \frac{U''}{U-c}\psi=0$$ where $\psi(y)$ is a complex valued function on $[0,2\pi]$ satisfying Dirichlet boundary conditions $\psi(0)=\psi(2\pi)=0$. $U(y)$ is a known, real valued function. $c$ is a complex number, $k$ is real.
This arises in fluid dynamics by linearizing the Euler equation of ideal fluid dynamics about a steady state $U(y)$ (the so called shear flow in a two dimensional channel) and considering a perturbation stream function of the form
$\hat{\psi}(x,y,t)=\psi(y) e^{ik(x-ct)}$
I have a few questions regarding this equation
i) It is claimed that this is an eigenvalue problem with $c$ as the complex eigenvalue. Why is this so? How does one rewrite this equation as an eigenvalue problem in $c$?
ii) Since the coefficients of the operator are real, eigenvalues occur in complex conjugate pairs. ( My suspicion is that they are talking about the operator $L\psi= \psi'' - k^2 \psi$)
iii) There exists a continuous spectrum for all (real) $c$ between $U_{min} \leq c \leq U_{max}$ with eigenfunctions that have a discontinuous derivative at $y_c$ where $y_c$ is the point such that $U(y_c)=c$