The Rayleigh hydrodynamical problem: a theorem on eigenfunction expansion and the stability of plane-parallel flows

The Rayleigh problem on the stability of plane-parallel flow of an ideal fluid leads to a singular and non-self-adjoint boundary-value problem that admits an operator formulation within the framework of the Friedrichs model. Using the technique of the stationary theory of scattering and the method of contour integration of the resolvent, a spectral analysis of the problem is carried out. The finiteness of the set of eigenvalues is proved, analytic properties of the Green's function are investigated, and the expansion in eigenfunctions corresponding to the continuous and point spectra is obtained. As an application, a time-asymptotic formula for the solution of the original non-stationary equation is derived.