Simplicity of the Second Dirichlet Eigenvalue on Triangles: Analytical Estimates and Computer-Assisted Proof

par Endo Ryoki (Durham University)

lundi 19 oct. 2026, 14:30 → 15:30 Europe/Paris

313 (Site des Carmes)

This talk presents a proof that the second eigenvalue of the Dirichlet Laplacian is simple

on every non-equilateral triangle, thereby resolving a conjecture of Laugesen and

Siudeja. The proof combines analytical estimates with verified numerical computation.

The principal difficulties arise near the equilateral configuration, where the second and

third eigenvalues coalesce, and in the nearly degenerate regime, where the eigenvalues

diverge. A difference-quotient formulation based on the Hadamard shape derivative

provides rigorous control of eigenvalue splitting near the equilateral triangle. For nearly

degenerate triangles, suitable eigenvalue bounds are obtained through comparison with

one-dimensional Schrödinger operators. Together with verified finite element computations,

these estimates establish simplicity for the entire class of non-equilateral triangles.

This is joint work with Xuefeng Liu.