Representing the smoothness of intermediate-eigenvalue solutions E(n, ε)

Develop a precise and explicit representation of the smoothness properties of eigenvalue solutions E(n, ε) for E ∈ [c_1 ε, c_2] that satisfy the intermediate-regime equation J(E) = (n + 1/2) ε + ε^{3/2} (E^{-1} ε)^{1/6} · \overline J_2((E^{-1} ε)^{1/3}, E^{1/2}, (-1)^n), clarifying their dependence on n and ε.

Background

The authors introduce an implicit equation for intermediate eigenvalues that bridges low-lying (E = O(ε)) and large (E = O(1)) regimes. They note that, by smooth change of variables, one can solve for E as a function of n and ε in [c_1 ε, c_2], overlapping with other regimes.

However, they explicitly state they have not found a good way to represent the smoothness properties (e.g., in ε and n) of these solutions. Providing such a representation would complete the global description of eigenvalue dependence across regimes.