Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.16399

An inverse problem on eigenfunction triple products

Carl Schildkraut·Romain Speciel
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.SPmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Carl Schildkraut, Romain Speciel. 2026-09-14. An inverse problem on eigenfunction triple products. https://arxiv.org/abs/2609.16399

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Sharp Neumann eigenvalue means on triangles:an exact certificate proof and stability

We prove both inequalities in Laugesen and Siudeja's Conjecture~6.34 for the first two positive Neumann eigenvalues of triangles. The equilateral triangle uniquely maximizes their harmonic mean at fixed perimeter and their geometric mean at fixed area. Two-dimensional trial spaces reduce the proof to polynomial positivity. Projected cubic corrections repair the second-order error near equality; elementary estimates and rational Bernstein certificates cover the remaining shapes. We provide exact data and a verifier, and derive an explicit quadratic stability estimate in terms of side-length asymmetry.

math.SP↗
arXiv2026

Hölder continuity and Fourier asymptotics of spectral measures for 1D Schrödinger operators under exponentially decaying perturbations

We establish $\frac{1}{2}$-Hölder continuity, or even the Lipschitz property, for the spectral measures of half-line discrete Schrödinger operators under suitable boundary conditions and sufficiently fast polynomially decaying small potentials. These are the first known examples, apart from the free case, of Schrödinger operators with Lipschitz continuous spectral measures up to the spectral edge, a property obtained here as a consequence of the Dirichlet boundary condition. Notably, we show that the asymptotic behavior of the time-averaged quantum return probability, either $\log (t) / t$ or $1 / t$, as in the case of the free Laplacian, remains unchanged in this setting. Furthermore, we prove the persistence of the purely absolutely continuous spectrum and the $\frac{1}{2}$-Hölder continuity of the spectral measures for (Diophantine) quasi-periodic operators under exponentially decaying small perturbations. These results are optimal and hold for all energies, up to the border of the absolutely continuous spectrum.

math.SP↗
arXiv2026

Bouncing-ball modes in ergodic billiards

We show that for a horizontally stretched Bunimovich billiard, for almost every stretching parameter, there exists a sequence of eigenfunctions concentrating in the rectangular part. Thanks to the work of Markarian--Oliffson Kamphorst--Pinto de Carvalho, we know that for an open set of parameters such billiards are ergodic. The proof is a result of interaction with ChatGPT 6: we were curious if it could establish the existence of bouncing ball modes for Bunimovich or Sinai billiards. That was not successful until we suggested varying the wings. That produced an essentially complete argument for the existence of quasimodes. We then suggested using Hassell's parameter-variation argument which led to the result about eigenfunctions. The argument was then significantly simplified and clarified by the authors.

math.SP↗