积分幂的自共轭弗里德里希-勒让德算子的新的特征化
A New Characterization of the Domains of Integral Powers of the Self-Adjoint Friedrichs-Legendre Operator
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中文总结 AI 辅助
本文提出了一种新的方法，用于特征化自共轭弗里德里希-勒让德算子的积分幂的定义域，证明其通过一个积分条件而非多个边界条件，并展示了光滑性结果的最优性。
中文摘要 AI 辅助
设A是生成由第二类经典勒让德微分方程生成的自共轭算子，该方程为：\[ \ell\lbrack y](t)=-\left( (1-t^{2})y^{\prime}(t)\right) ^{\prime}+ky(t)=\lambda y(t)\quad(t\in(-1,1)), \] 其勒让德多项式$\{P_{m}\}_{m=0}^{\infty}$构成完整的特征函数序列；这里k是一个固定的非负实数。这是与$\ell[\cdot]$在$L^2(-1,1)$中关联的最小算子的弗里德里希扩展。对于每个$n \in \mathbb{N}$，我们证明$\mathcal{D}(A^{n})$可以通过一个积分可积条件来特征化，而不是由经典格拉兹曼-克雷因-奈马克理论所规定的2n个边界条件。我们还证明如果$f\in\mathcal{D}(A^{n})$，则$f^{(n)}\in L^{2}(-1,1)$。这个光滑性结果在$n=1$和$n=2$时扩展了已知的结果。此外，这一结果在最优意义上成立，即存在$g\in\mathcal{D}(A^{n})$使得$g^{(n+1)}\notin L^{2}(-1,1)$。
英文摘要
Let $A$ be the self-adjoint operator in $L^{2}(-1,1)$, generated by the second-order classical Legendre differential equation% \[ \ell\lbrack y](t)=-\left( (1-t^{2})y^{\prime}(t)\right) ^{\prime}+ky(t)=λy(t)\quad(t\in(-1,1)), \] which has the Legendre polynomials $\{P_{m}\}_{m=0}^{\infty}$ as a complete sequence of eigenfunctions; here $k$ is a fixed, non-negative real number. This is the Friedrichs extension of the minimal operator associated with $\ell[\cdot]$ in $L^2(-1,1)$. For each $n \in \mathbb{N}$, we show that $\mathcal{D}(A^{n})$ is characterized by \textit{one} integrability condition instead of $2n$ boundary conditions as dictated by the classical Glazman-Krein-Naimark theory. We also prove that if $f\in\mathcal{D}(A^{n})$ then $f^{(n)}\in L^{2}(-1,1).$ This smoothness result extends known results when $n=1$ and $n=2.$ Furthermore, this result is optimal in the sense that there exists $g\in\mathcal{D}(A^{n})$ with $g^{(n+1)}\notin L^{2}(-1,1)$.
