Maximizers of λk/λ1 on Dirichlet trees

Determine, for each fixed integer k≥2, the compact Dirichlet trees that maximize the eigenvalue ratio λk(Γ)/λ1(Γ); in particular, ascertain whether the interval maximizes λ3(Γ)/λ1(Γ), and investigate whether the interval maximizes λk(Γ)/λ1(Γ) for k≥4.

Background

The paper proves sharp ratio bounds for Dirichlet trees, including a universal Ashbaugh–Benguria-type bound for λ2/λ1 and general bounds λk/λj, and characterizes equality for λ2/λ1. However, the authors do not settle which trees maximize λk/λ1 for higher k.

They note an analogy with classical shape optimization on domains, where intervals (1D balls) are often extremizers, and report numerical evidence suggesting the interval might maximize λk/λ1 even for k≥4. Establishing global maximizers across all Dirichlet trees remains open.

References

Open Problem. Study the problem of maximizing the ratios \frac{\lambda_k}{\lambda_1} on Dirichlet trees. In particular, as a 1-dimensional analogue of [Open problem 14] we can expect that \frac{\lambda_3}{\lambda_1} should be maximized by the interval. (See Remark~\ref{rem:big-ab}(3).) Numerical evidence suggests that even for general k \geq 4, \frac{\lambda_k}{\lambda_1} might attain its maximum on the interval.

— Bounds on eigenvalue ratios of quantum graphs  (2603.26172 - Harrell et al., 27 Mar 2026) in Open Problem, end of Introduction (Section 1)