Finite inverse nodal problems for singular weighted Sturm--Liouville equations: variational selection and spectral matching
Abstract
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)=rn−1, rather than to a regular interval model. We minimize the PDE distance |Q−Q0|Lp(BR) , equivalently the weighted distance |q−q0|Lpw(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 C1 manifold structure. Crucially, without the variational selection imposed by q0, 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.
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BibTeX
@misc{cheng2026finite,
title = {{Finite inverse nodal problems for singular weighted Sturm--Liouville equations: variational selection and spectral matching}},
author = {Zhibo Cheng and Yuchao He and Sunan Wang and Yonghui Xia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16481},
eprint = {2609.16481},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16481},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Finite inverse nodal problems for singular weighted Sturm--Liouville equations: variational selection and spectral matching
AU - Zhibo Cheng
AU - Yuchao He
AU - Sunan Wang
AU - Yonghui Xia
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.16481
UR - https://arxiv.org/abs/2609.16481
AB - 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.
ER -