Reducibility of quantum harmonic oscillator on R^d perturbed by a quasi-periodic potential with logarithmic decay

The paper proves reducibility for the quantum harmonic oscillator on R^d with a quasi-periodic-in-time potential that decays logarithmically in space, under iota ≥ (n+d+1)/2, and obtains the quantitative bounds meas(D0\Dε) ≤ C ε^{1/6}, ‖W‖_{B(H^s)} ≤ Cε, and ‖Ψ^{±1}−Id‖_{B(H^p)} ≤ C ε^{5/12} for p∈[0,1]. The candidate solution reproduces the same KAM reducibility scheme in the Hermite basis, the same hypotheses and end-state (unitary, analytic conjugacy Ψ; block-diagonal W commuting with H0; and the same exponents). Minor presentational differences (e.g., how the matrix class and small-divisor conditions are described) do not affect correctness. Overall, the argument paths are essentially the same.