Absence and explanation of the ε^{5/3} correction term in the Bohr–Sommerfeld approximation

Determine whether the ε^{5/3} correction term in the large-eigenvalue relation J(E_n) = (n + 1/2) ε + ε^{5/3} · overline J(n ε, ε^{1/3}, (-1)^n) derived in Theorem 1 (theorem:#1{01eigenvalues}) actually vanishes, thereby yielding an O(ε^2) remainder consistent with Yafaev (2011), and provide a rigorous explanation reconciling the ε^{5/3} remainder obtained via the dynamical-systems approach with the O(ε^2) remainder in WKB analyses.

Background

The authors prove that, under smooth single-well assumptions, the Bohr–Sommerfeld quantization formula is perturbed by a term of order ε^{5/3} with smooth dependence on nε and ε^{1/3}. Classical WKB results (e.g., Yafaev 2011, Theorem 4.1) assert an O(ε^2) remainder for the Bohr–Sommerfeld approximation.

For the two approaches to agree, the ε^{5/3} term would need to be absent; the authors explicitly state they have neither verified this nor found a direct explanation. Establishing the absence of the ε^{5/3} term and reconciling the discrepancy would close a gap between these frameworks.