Finite-data inverse nodal optimization in angular-momentum sectors of Schrödinger operators
Abstract
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 Ldp, 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 ℓ=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 ℓ=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.
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BibTeX
@misc{deng2026finitedata,
title = {{Finite-data inverse nodal optimization in angular-momentum sectors of Schrödinger operators}},
author = {Xijun Deng and Zhisu Liu and Yonghui Xia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23502},
eprint = {2609.23502},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23502},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Finite-data inverse nodal optimization in angular-momentum sectors of Schrödinger operators
AU - Xijun Deng
AU - Zhisu Liu
AU - Yonghui Xia
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.23502
UR - https://arxiv.org/abs/2609.23502
AB - 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 ℓ=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 ℓ=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.
ER -