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In this paper, for an operator defined by the action of an M-th orderdifferential operator with rational-type coefficients on the function spaceL_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 anddifferentiability up to M times) of the eigenfunctions of its closure (withrespect to the graph norm), except at singular points of the correspondingordinary differential equation without any assumptions for the Sobolev space,i.e., without any assumptions about the m-th order derivatives of theeigenfunctions with m=1,2,.., M-1. (For the special case of k=0, we prove thisregularity for the usual L^2(R).) Especially, we show a one-to-onecorrespondence between the eigenfunctions of its closure and the solutions inC^M(R)\cap L_k^2(R) of the corresponding differential equation under thecondition above when there is no singular point for this differential equation.This one-to-one correspondence is shown in the basic framework of an algorithmproposed in our preceding paper, which can determine all solutions in C^M\capL_k^2(R) of the ordinary differential equation then.

Differentiability of eigenfunctions of the closures of differential operators with rational coefficient functions