奇异加权Sturm--Liouville方程的有限逆节点问题：变分选择与谱匹配

中文摘要 AI 辅助

近期工作\\(\cite{HeWuXiaZhang2025}\\)为正则一维Sturm-Liouville算子引入了有限数据逆节点框架。然而，为Schrödinger算子建立多维逆节点理论仍是一个具有挑战性的开放问题。本文通过为球中Schrödinger算子的径向谱分支建立有限径向逆节点理论来填补这一空白。与区间情形不同，节点数据是径向本征函数的球面节点超曲面的半径。精确的径向约化保留了这种PDE节点几何结构，并自然地导出一个具有权重\\(w(r)=r^{n-1}\\)的奇异加权Sturm-Liouville问题，而非正则区间模型。我们在所有匹配有限多个给定球面节点半径的径向势中，最小化PDE距离\\(\\|Q-Q_0\\|_{L^p(B_R)}\\)，等价于加权距离\\(\\|q-q_0\\|_{L^p_w(0,R)}\\)。对于相关的节点约束集，我们证明了谱正则性、节点可微性、弱闭性、精确实现性以及有限余维\\(C^1\\)流形结构。关键的是，如果没有由\\(q_0\\)施加的变分选择，给定的有限节点半径无法唯一确定势：可容许势在常数平移下不变，且有限节点约束在正则点处产生局部无限维水平集。

英文摘要

The recent work \cite{HeWuXiaZhang2025} introduced a finite-data inverse nodal framework for regular one-dimensional Sturm-Liouville operators. Nevertheless, formulating a multidimensional inverse nodal theory for Schrödinger operators remains a challenging open problem. This paper addresses this gap by establishing a finite radial inverse nodal theory for the radial spectral branch of Schrödinger operators in balls. In contrast with the interval case, the nodal data are the radii of spherical nodal hypersurfaces of radial eigenfunctions. The exact radial reduction preserves this PDE nodal geometry and leads naturally to a singular weighted Sturm--Liouville problem with weight $w(r)=r^{n-1}$, rather than to a regular interval model. We minimize the PDE distance $\|Q-Q_0\|_{L^p(B_R)}$ , equivalently the weighted distance $\|q-q_0\|_{L^p_w(0,R)}$, among all radial potentials matching finitely many prescribed spherical nodal radii. For the associated nodal constraint sets, we prove spectral regularity, nodal differentiability, weak closedness, exact realization, and a finite-codimensional $C^1$ manifold structure. Crucially, without the variational selection imposed by \(q_0\), the prescribed finite nodal radii fail to determine the potential uniquely: the admissible potentials are invariant under constant shifts, and the finite nodal constraints yield locally infinite-dimensional level sets at regular points.

发表机构

• School of Mathematica and Information Science, Henan Polytechnic University（河南理工大学数学与信息科学学院）
• School of Mathematical Science, Zhejiang Normal University（浙江师范大学数学系）
• School of Mathematics, Foshan University（佛山大学数学学院）

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