Semiclassical formulae for Wigner distributions

The uploaded paper’s Theorem 4.1 states precisely that for closed constant-curvature surfaces, first-band Ruelle resonances λ with Re λ > −1/2 correspond to Laplace–Beltrami eigenvalues E = (iλ)^2 + 1/4, and that for purely imaginary λ = ir → i∞ the associated invariant Ruelle distribution T_λ(a) equals the sum of quantum expectations over the eigenspace at E = 1/4 + r^2 up to O(1/r) for fixed a. This matches the candidate solution’s parts (a) and (b) verbatim. The paper also recalls the meromorphic continuation of the resolvent and the interpretation of residues as invariant Ruelle distributions, supporting the model’s Step (1). Minor bibliographic emphasis differs (the O(1/r) comparison is attributed in the paper to GHW21 based on AZ07), but there is no mathematical conflict. Overall, both are correct and aligned in substance, with the model giving a compatible proof sketch using the same underlying machinery. See Theorem 4.1 and the surrounding discussion in Section 4, as well as the semiclassical–Ruelle residue correspondence in Section 3 and the resolvent meromorphy in Section 2 of the uploaded paper .