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

Corentin Léna (1)

1 Dipartimento di Matematica "Giuseppe Peano" [Torino]

In this note, we discuss a question posed by T. Hoffmann-Ostenhof (see [3]) 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.

Mots clés: nodal domains; torus; Laplacian

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]; Théorie spectrale [math.SP]

Corentin Léna. On the parity of the number of nodal domains for an eigenfunction of the Laplacian on tori. 2015. ⟨hal-01140911⟩