Recovering the angles of a curvilinear polygon from its Steklov spectrum
Let $\Omega \subset \mathbb R^2$ be a curvilinear polygon with side lengths $\ell_1, \dots, \ell_n$ in order and interior angles $\alpha_1, \dots, \alpha_n$, all in $(0, \pi)$ and none of the form $\pi/m$ with $m \in \mathbb N$, and with the lengths incommensurable over $\{-1, 0, 1\}$. Krymski, Levitin, Parnovski, Polterovich and Sher proved that the Steklov spectrum of such a polygon determines, constructively, the side lengths in order (up to cyclic relabelling and reversal of orientation) and, up to a global sign, the vector $\big(\cos\frac{\pi^2}{2\alpha_1}, \dots, \cos\frac{\pi^2}{2\alpha_n}\big)$. The spectral asymptotics do not distinguish two angles with the same value of $\cos(\pi^2/2\alpha)$, so recovering the angles needs non-asymptotic information from the spectrum.
Problem. Under these hypotheses, can the angles $\alpha_1, \dots, \alpha_n$ themselves be recovered from the Steklov spectrum?
Source: Posed by Colbois, Girouard, Gordon and Sher (2024), Open Question 8.14, after Krymski, Levitin, Parnovski, Polterovich and Sher (2021).
The statement is adapted from doi.org/10.1007/s13163-023-00480-3, whose text is available under CC BY 4.0.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP
- Keywords
- Colbois–Girouard–Gordon–Sher Steklov problems, inverse spectral problems, polygons, Steklov eigenvalues
References
@article{krymski2021,
author = {Krymski, Stanislav and Levitin, Michael and Parnovski, Leonid and Polterovich, Iosif and Sher, David A.},
title = {Inverse {S}teklov spectral problem for curvilinear polygons},
journal = {International Mathematics Research Notices},
volume = {2021},
number = {1},
pages = {1--37},
year = {2021},
doi = {10.1093/imrn/rnaa200},
eprint = {2004.03881},
archivePrefix = {arXiv}
}
@article{dryden2025,
author = {Dryden, Emily B. and Gordon, Carolyn and Moreno, Javier and Rowlett, Julie and Villegas-Blas, Carlos},
title = {The {S}teklov spectrum of convex polygonal domains {I}: spectral finiteness},
journal = {Journal of Geometric Analysis},
volume = {35},
pages = {Paper No. 91},
year = {2025},
doi = {10.1007/s12220-025-01922-8},
eprint = {2408.01529},
archivePrefix = {arXiv}
}
@misc{dryden2026,
author = {Dryden, Emily B. and Gordon, Carolyn and Moreno, Javier and Rowlett, Julie and Villegas-Blas, Carlos},
title = {The {S}teklov spectrum of convex polygonal domains {II}: investigating spectral determination},
year = {2026},
eprint = {2604.18977},
archivePrefix = {arXiv}
}
@article{colbois2024,
author = {Colbois, Bruno and Girouard, Alexandre and Gordon, Carolyn and Sher, David},
title = {Some recent developments on the {S}teklov eigenvalue problem},
journal = {Revista Matem{\'a}tica Complutense},
volume = {37},
number = {1},
pages = {1--161},
year = {2024},
doi = {10.1007/s13163-023-00480-3},
eprint = {2212.12528},
archivePrefix = {arXiv},
note = {Open Question 8.14}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01027, "Recovering the angles of a curvilinear polygon from its Steklov spectrum", clopen.solutions.