Improving smoothness from ε^{1/3} to ε for large-eigenvalue corrections

Ascertain whether the correction function overline J(n ε, ε^{1/3}, (-1)^n) in the large-eigenvalue formula J(E_n) = (n + 1/2) ε + ε^{5/3} · overline J(n ε, ε^{1/3}, (-1)^n) can be reformulated to depend smoothly on ε (without fractional powers), and, if feasible, construct such a representation under assumptions (A), (smoothness), and (growth).

Background

A central novelty of the paper is that the correction terms for large eigenvalues depend smoothly on ε^{1/3}, reflecting turning-point behavior captured by the authors’ dynamical-systems method.

The authors remark that the ε^{1/3} dependence might be method-induced and explicitly state they have not found a way to improve the smoothness to ε. Establishing such an improvement would refine the asymptotic description and align it with smoother parameter dependence.