薛定谔算子角动量扇区中的有限数据逆节点优化
Finite-data inverse nodal optimization in angular-momentum sectors of Schrödinger operators
- Hubei University of Automotive Technology（湖北汽车工业学院）
- China University of Geosciences（中国地质大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对径向薛定谔算子在角动量扇区中的有限数据逆节点优化问题，证明了节点半径的连续可微性、最小距离势的存在性及临界方程，并给出局部重构与唯一性结果。
AI中文摘要：
本文研究了球B_R⊂R^d（d≥2）上径向薛定谔算子H_q:=-Δ+q(|x|)，u|_∂B_R=0在任意固定角动量扇区中的有限数据逆节点优化问题。分析直接在Friedrichs端点以及物理加权空间L_d^p（p>d/2）中进行，从而保留了奇异径向几何而非用正则一维模型替代。通过Friedrichs分支的Volterra表示，我们建立了固定节点半径关于势的弱连续性和连续Fréchet可微性。然后我们证明了最小距离势的存在性并推导了相应的临界薛定谔方程。对于同一本征函数的多个节点，节点梯度线性无关，这产生了有限维浸没结构以及尖锐的局部最小范数重构。进一步利用一般观测原理处理混合角动量数据（在相应的横截条件下）以及来自单个本征模式的同步谱-节点数据（其横截性自动满足）；在Hilbert情形下，这给出了显式的逆Gram公式和局部唯一性。最后，当ℓ=0，参考势为常数，并考虑第二径向模式时，我们证明对于p>(d+2)/2，唯一内部节点的每个向内位移都拥有唯一的全局优化子。
英文摘要：
In this paper, we study a finite-data inverse nodal optimization problem for radial Schrödinger operators in an arbitrary fixed angular-momentum sector. The analysis is built directly at the Friedrichs endpoint and in the physical weighted space $L_d^p$, so that the singular radial geometry is retained rather than replaced by a regular one-dimensional model. The main purpose of this paper is to provide \emph{a singular Friedrichs finite-data variational framework} valid in every angular-momentum sector, thereby extending the existing finite-data variational theories concerning either regular one-dimensional operators or the radial sector $\ell=0$. By means of a Volterra representation of the Friedrichs branch, we prove weak continuity and continuous Fréchet differentiability of nodal radii, exact realization of compatible same-mode nodal data, existence of optimal potentials, and finite-codimensional constraint geometry. The same framework also incorporates mixed angular-momentum and spectral--nodal observations through finite-dimensional transversality. Remarkably, \emph{a global uniqueness theorem} is established for inward displacements of the unique interior node of the second mode in the radial sector \(\ell=0\). For a constant reference potential and \(p>(d+2)/2\), every such displacement admits a unique global optimizer. Unlike local inverse-mapping or one-dimensional integrability arguments, the proof first selects the admissible critical sign globally and then reduces every minimizer to a scalar mass-balance equation between a focusing ball branch and a logistic annulus branch. The strict opposite monotonicity of the two weighted masses makes the balance parameter unique, providing a global rigidity mechanism over the entire inward-displacement regime.
