Expansions in terms of sets of functions with complex eigenvalues

In the following I discuss the properties, in particular the completeness of the set of eigenfunctions, of an eigenvalue problem which differs from the well-known Sturm-Liouville problem by the boundary condition being of a rather unusual type. The problem arises in the theory of nuclear collisions, and for our present purpose we take it in the simplified form where 0 ≤ x ≤ 1. V ( x ) is a given real function, which we assume to be integrable and to remain between the bounds ± M , and W is an eigenvalue. The eigenfunction ψ( x ) is subject to the boundary conditions and