Abstract :
Left definite theory of regular self-adjoint operators in a Sobolev space was developed a few years ago when the boundary conditions were separated, the separation being necessary in order to properly define the Sobolev inner product. We show how this may be extended when the boundary conditions are not separated, but when evaluations at both ends are mixed together. There are essentially three cases which arise: First when a coefficient matrix is nonsingular, second when it is singular but not 0, third when it is 0. The middle case does not arise under separated conditions.
