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An inverse problem on eigenfunction triple products
Abstract
On a connected closed smooth Riemannian manifold, the algebraic structure of the Laplace eigenfunctions, as described by eigenfunction triple products, uniquely determines the geometry. We refine this correspondence by introducing the notion of an N-product eigenbasis, which consists of eigenfunctions whose pairwise products may be written as linear combinations of at most N basis elements. We prove that a manifold admits a $2$-product eigenbasis if and only if it is a flat torus. We also prove an analogous result for Laplace eigenvectors of bounded-degree graphs.
Related
- Inferring geometric structure from eigenfunction triple products
- Transposed Triple Products and Pro-Symmetric Rings in ∗-Rings
- The embedded Nash problem of birational models of rational triple singularities
- Triple Products of Correlated Multivariate Hermite Polynomials
- KS-Groupoids and KS-Crossed Products associated with Inverse Semigroup Actions
BibTeX
@misc{schildkraut2026inverse,
title = {{An inverse problem on eigenfunction triple products}},
author = {Carl Schildkraut and Romain Speciel},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16399},
eprint = {2609.16399},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16399},
}
RIS
TY - UNPB
TI - An inverse problem on eigenfunction triple products
AU - Carl Schildkraut
AU - Romain Speciel
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.16399
UR - https://arxiv.org/abs/2609.16399
AB - On a connected closed smooth Riemannian manifold, the algebraic structure of the Laplace eigenfunctions, as described by eigenfunction triple products, uniquely determines the geometry. We refine this correspondence by introducing the notion of an N-product eigenbasis, which consists of eigenfunctions whose pairwise products may be written as linear combinations of at most N basis elements. We prove that a manifold admits a $2$-product eigenbasis if and only if it is a flat torus. We also prove an analogous result for Laplace eigenvectors of bounded-degree graphs.
ER -