Implicit ODE characterization of the eigenvalue curve

Derive an implicit characterization of the curve \(k\mapsto\lambda_{k,\alpha k}\), possibly by identifying an ordinary differential equation whose solution determines the first radial \((k,\alpha k)\)-eigenvalues.

Background

The paper determines the asymptotic limit of λk,n\lambda_{k,n} when n/kn/k tends to a fixed ratio and proves monotonicity for the normalized eigenvalues. It does not provide an implicit description of the full finite-kk curve k↦λk,αkk\mapsto\lambda_{k,\alpha k}. The authors explicitly raise the possibility of obtaining such a description through another ODE.

References

Finally, it is additionally interesting to ask if one can determine the curve $k \mapsto \lambda_{k,\alpha k}$ implicitly, perhaps as the solution to another ODE.

— Asymptotics for the $k$-Hessian Eigenvalue on the Unit Ball  (2609.05277 - McCleerey et al., 4 Sep 2026) in Section 7, Numerical Results and Further Questions