Courant-sharp eigenvalues of a two-dimensional torus

TL;DR

This paper characterizes all Courant-sharp eigenvalues of the Laplacian on a flat two-dimensional torus, combining Weyl's law, Faber-Krahn inequality, and explicit bounds to identify these special eigenvalues.

Contribution

It provides a complete classification of Courant-sharp eigenvalues for the Laplacian on the 2D torus, extending previous partial results with explicit bounds and inequalities.

Findings

Identifies all Courant-sharp eigenvalues on the 2D torus

Establishes explicit bounds using Weyl's law and Faber-Krahn inequality

Demonstrates the finiteness of Courant-sharp eigenvalues in this setting

Abstract

In this paper, we determine, in the case of the Laplacian on the flat two-dimensional torus (R/Z) 2 , all the eigenvalues having an eigenfunction which satisfies Courant's theorem with equality (Courant-sharp situation). Following the strategy o A. Pleijel (1956), the proof is a combination of a lower bound a la Weyl) of the counting function, with an explicit remainder term, and of a Faber--Krahn inequality for domains on the torus (deduced as in B{\'e}rard-Meyer from an isoperimetric inequality), with an explicit upper bound on the area.