Occurrence of geometrically double Orr–Sommerfeld eigenvalues

Determine whether the Orr–Sommerfeld operator for the linearized two-dimensional plane Poiseuille channel ever has an eigenvalue with geometric multiplicity two, equivalently whether the hypothesis of Proposition 2.2 is realized for any wavenumber and Reynolds number.

Background

Proposition 2.2 shows that if an Orr–Sommerfeld eigenvalue has a two-dimensional eigenspace, then feedback through either the tangential or normal wall-velocity component alone cannot move that eigenvalue; in particular, a linearly unstable eigenvalue of this type cannot be stabilized using only one component. The authors note that the Orr–Sommerfeld operator is symmetric about the channel midline, so its eigenfunctions split into even and odd families, and a coincidence between the two families could produce geometric multiplicity two.

The paper’s cited degeneracies instead involve eigenvalue coalescences producing a single mode of algebraic order two, rather than a two-dimensional eigenspace. The proposed two-component design does not require resolving whether geometrically double eigenvalues actually occur, because it handles arbitrary multiplicities and Jordan chains.