  In this paper, for an operator defined by the action of an M-th order
differential operator with rational-type coefficients on the function space
L_k^2(R):={f: measurable | \|f\|_k <\infty} with norm \|f\|_k^2:= \int |f(x)|^2
(x^2+1)^k dx (k \in Z), we prove the regularity (continuity and
differentiability up to M times) of the eigenfunctions of its closure (with
respect to the graph norm), except at singular points of the corresponding
ordinary differential equation without any assumptions for the Sobolev space,
i.e., without any assumptions about the m-th order derivatives of the
eigenfunctions with m=1,2,.., M-1. (For the special case of k=0, we prove this
regularity for the usual L^2(R).) Especially, we show a one-to-one
correspondence between the eigenfunctions of its closure and the solutions in
C^M(R)\cap L_k^2(R) of the corresponding differential equation under the
condition above when there is no singular point for this differential equation.
This one-to-one correspondence is shown in the basic framework of an algorithm
proposed in our preceding paper, which can determine all solutions in C^M\cap
L_k^2(R) of the ordinary differential equation then.
