On singular ordinary linear differential operators related to Orr-Sommerfeld stability equation

A usable, not technically complicated theory unlike that of Wasow of linear, singular, even order differential operators with general boundary conditions is proposed and the behaviour of eigenvalues and eigenfunctions is discussed in the limit as ε → 0, ε > 0. Completeness of the eigenvalue spectrum when the operator is formally self-adjoint is discussed and formal solutions are constructed and finally justification of the formal approximations is shown by proving the existence of actual solutions of the differential equation approximated by formal solutions. Also, modifications necessary for the consideration of the critical point are suggested. This theory is unlike that of Reid since it provides rigorous justification of the approximations and moreover, this theory is applicable to a large class of physical problems rewritten in such a way that they contain a small parameter.