Eigenfunction expansions and non-selfadjoint boundary value problems

Abstract

We shall consider a non-selfadjoint boundary value problem for (1) (Δ + k²)u = F. In Section 2 we apply well-known techniques [1]-[5] to obtain representations for the solution u(r, θ) in polar coordinates. These techniques assume the existence of a certain eigenfunction expansion involving the eigenfunctions of an ordinary differential system which results on applying separation of an ordinary differential system which results on applying separation of variables to (1). However, in Section 3 we show that for a certain class of (physically interesting) functions such an expansion does not exist. Pflumm [6] seems to have been the first to have noticed this. We shall state his results in Section 3. Furthermore, we show that even the solution of the ordinary differential system cannot be expanded in a series of these eigenfunctions. Nevertheless, our main result (Section 4) is that for a certain class of functions F the solution u of the partial differential equation (1) can be expanded in a convergent series of these eigenfunctions, but only in a subdomain of the original domain where a solution to (1) exists. Thus, the representation of the solution to (1), obtained formally in Section 2, is valid in a certain subdomain even though the method of the derivation cannot be justified. However, it is not possible to find the region of validity of the representation from the formal derivation.