On the parity of the number of nodal domains for an eigenfunction of the Laplacian on tori

2015/04/15 by Corentin Léna, Léna, Corentin
Mathematics · Physics and Astronomy · #35J05 #35P99 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometry and complex manifolds #Nonlinear Partial Differential Equations #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.AP #math.SP #msc:35J05 #msc:35P99

paper · pdf · doi:10.48550/arxiv.1504.03944

5 pages, 2 figures

openalex publication_date 2015/04/15 · arxiv created 2015/07/14 · arxiv updated 2015/07/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note, we discuss a question posed by T. Hoffmann-Ostenhof concerning the parity of the number of nodal domains for a non-constant eigenfunction of the Laplacian on flat tori. We present two results. We first show that on the torus (ℝ/2πℤ)², a non-constant eigenfunction has an even number of nodal domains. We then consider the torus (ℝ/2πℤ)×(ℝ/2ρπℤ) , with ρ=(1)/(√(3)) , and construct on it an eigenfunction with three nodal domains.

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