On the reducibility of the 1d quantum harmonic oscillator with a quasi-periodic bounded potential

The uploaded paper proves reducibility for i∂tψ = (−∂2x + x2)ψ + εV(x,ωt)ψ under |V|≤C and |∂xV|≤C(1+|x|)^δ with δ∈(0,1), yielding a unitary, real-analytic conjugation to a diagonal autonomous system, with |λ∞i−λi|≲ε, ||Ψ±−id||≲ε2/3, and Meas(Π\Πε)≲ε^{(1−δ)/(17(5−δ))} (Theorem 1.1) . The candidate solution reproduces the same target bounds via a KAM scheme built on Hermite-basis diagonal smoothing from the commutator identity (their bound matches Lemma 2.6’s 2|i−j||Pji|≲max{i,j}^{1/2}(i∧j)^{δ/2}) and the matrix formulation iu̇=(A+εP)u with A=diag{1,3,5,…} and Pj i(t)=∫Vhi hj dx . Technically, the paper controls homological equation solutions using the difference-matrix space M̂α and obtains ||Ŝ(k)||_{α̂+}≲(K/κ2)||P̂(k)||_{α̂}, with a small-divisor set given by |k·ω+Λi−Λj|≥κ(1+|i−j|) and measure Meas(D\D′)≲κ^{α/(α+2)}K^{n+1} (Proposition 3.1) . By contrast, the candidate sketches a more standard Diophantine/Toeplitz–Lipschitz route, introducing an ℓ^τ loss and an additional near-diagonal cut-off. Despite these stylistic differences, both arguments hinge on the same Hermite-diagonal smoothing mechanism and reach the identical quantitative conclusions stated in the paper’s main theorem.