Differentiability of Eigenfunctions of the Closures of Differential Operators with Polynomial-type Coefficients

In this paper, for an operator defined by the action of an M -th order differential operator with polynomial-type coefficients on the function space L2 (k) (R) := {f : measurable|‖f‖(k) < ∞} with norm ‖f‖(k) = R |f(x)|2(x2 + 1)dx (k0 ∈ Z), we prove regularity (continuity and differentiability up to M times) of the eigenfunctions of its closure (with respect to the graph norm) under the condition that the coefficient polynomial of the highest-order term has no zero point, 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 L2(R).) Our main purpose is to show a one-to-one correspondence between the eigenfunctions of its closure and the solutions in C (R) ∩ L2 (k) (R) of the corresponding linear ordinary differential equation under the condition above. This one-to-one correspondence can be shown in the basic framework of an algorithm proposed in our preceding paper, which can determine all solutions in C ∩L2 (k) (R) of the ordinary differential equation. In this framework, the differential operator is treated as an operator from a Hilbert space H to another Hilbert space H , and it can be represented in matrix form with appropriate basis systems of H and H . In accordance with this matrix representation, we transform eigenfunctions in H to square-summable number sequences satisfying the matrix-vector equation. The truncation of this square-summable number sequence yields an appropriate approximation of the eigenfunction by an M -times differentiable function. We can show that the eigenfunction belongs to C (R) when the pair of Hilbert spaces H and H satisfies several conditions and this approximation has point-wise convergence.