CourantSharp Eigenvalues for the Equilateral Torus, and for the Equilateral Triangle
Abstract
We address the question of determining the eigenvalues {λ_{n}} (listed in nondecreasing order, with multiplicities) for which Courant's nodal domain theorem is sharp i.e., for which there exists an associated eigenfunction with {n} nodal domains (Courantsharp eigenvalues). Following ideas going back to Pleijel (1956), we prove that the only Courantsharp eigenvalues of the flat equilateral torus are the first and second, and that the only Courantsharp Dirichlet eigenvalues of the equilateral triangle are the first, second, and fourth eigenvalues. In the last section we sketch similar results for the rightangled isosceles triangle and for the hemiequilateral triangle.
 Publication:

Letters in Mathematical Physics
 Pub Date:
 December 2016
 DOI:
 10.1007/s1100501608199
 arXiv:
 arXiv:1503.00117
 Bibcode:
 2016LMaPh.106.1729B
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics;
 Mathematics  Differential Geometry;
 Mathematics  Spectral Theory
 EPrint:
 Slight modifications and some misprints corrected