Let T be a bounded triangle in the Euclidean plane, and let the Dirichlet eigenvalues of its Laplacian, listed with multiplicity, be 0<lambda_1(T)≤lambda_2(T)≤…. An equilateral triangle is a triangle whose three sides have equal length. Laugesen–Siudeja's conjecture. The second Dirichlet eigenvalue is simple on every non-equilateral triangle; equivalently, if T is not equilateral, then lambda_2(T)<lambda_3(T). This conjecture is solved by the paper, which gives a computer-assisted proof and in particular resolves the previously open case of nearly degenerate triangles.

Laugesen–Siudeja's simplicity conjecture for the second Dirichlet eigenvalue

The Laugesen–Siudeja conjecture asserts that every non-equilateral triangle satisfies $\lambda_2(T)<\lambda_3(T)$ , so its second Dirichlet eigenvalue is simple. The conjecture is now addressed by a claimed computer-assisted proof covering the previously unresolved nearly degenerate cases.

Known results

An earlier companion result handled the complementary family of triangles with minimum normalized height at least $\tan(\pi/60)/2$ ; the new work treats the remaining collapsing regime.

March 9, 2025 claimed complete solution

On March 9, 2025, Ryoki Endo and Xuefeng Liu’s preprint The second Dirichlet eigenvalue over any non-equilateral triangle is simple claimed the missing estimate for triangles with minimum normalized height at most $\tan(\pi/60)/2$ . Combined with the earlier result, it claims $\lambda_2(T)<\lambda_3(T)$ for every non-equilateral triangle; the proof is computer-assisted and remains unverified here.

Current status (as of September 2026): A March 2025 preprint claims the conjecture is completely solved, but the claim remains unverified; no counterexample or public correction was found.

Laugesen–Siudeja's simplicity conjecture for the second Dirichlet eigenvalue

Primary source

Progress summary

Solutions 0