The Spectral Concentration for Damped Waves on Compact Anosov Manifolds

The paper proves Theorem 1.3: in the O(h) window around Re z=1/2, the number of poles whose imaginary parts deviate from q by at least w(h) is O(h^{1−d} e^{c w(h)^2 |log h|} w(h)^{−2}), with c(q,M)=1/(2Λ0σ_q^2). The argument combines an Ehrenfest-time averaging/conjugation à la Sjöstrand–Anantharaman, a trace-class perturbation/determinant construction, Jensen counting on an h-dependent domain, and a moderate-deviation estimate for Birkhoff averages under Anosov flow; see (1.8)–(1.11), Proposition 3.1, and the counting step (5.4)–(5.8) culminating in Theorem 1.3 . The candidate solution follows the same blueprint: Egorov up to Ehrenfest time to replace Q by Op_h(⟨q⟩_T), a holomorphic determinant/Jensen argument reducing counting to phase-space volumes, and small-deviation (moderate) large-deviation bounds giving the e^{c w(h)^2|log h|} factor. It differs only in presentational details (e.g., not spelling out the h-dependent conformal mapping of the Jensen domain used in the paper), but the logic, scale w(h) ≫ |log h|^{-1/2}, and the constant c(q,M)=1/(2Λ0σ_q^2) all agree with the paper’s proof .