具有有理系数函数的微分算子闭包的特征函数的可微性
Differentiability of eigenfunctions of the closures of differential operators with rational coefficient functions
摘要：
本文证明了在函数空间L_k^2（R）上由有理型系数的M阶微分算子的作用定义的算子：={f：可测的|\|f\|_k < \infty }，范数为\|f\|_k^2：= \int| f（x）|^2（x^2+1）^kdx（k \in Z），我们证明了其闭包的特征函数（关于图范数）的正则性（连续性和M次可微性），除了相应的常微分方程的奇点，而没有对Sobolev空间作任何假设，即，在没有关于特征函数的m阶导数的任何假设的情况下，m= 1，2，..，M-1. (For在k=0的特殊情况下，我们证明了通常的L^2（R）的正则性。）特别地，当该微分方程不存在奇点时，在上述条件下，我们证明了其闭包的本征函数与相应的微分方程在C^M（R）\cap L_k^2（R）中的解之间的一一对应关系.这种一一对应在我们前一篇文章中提出的一个算法的基本框架中得到了证明，该算法可以确定常微分方程在C^M \cap L_k^2（R）中的所有解。
英文摘要：
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.
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