KAM for high-dimensional nonlinear quantum harmonic oscillator

The paper proves an abstract infinite-dimensional KAM theorem tailored to multiple normal frequencies with a weakened block-decay norm and applies it to the high-dimensional NLS with harmonic potential to produce quasi-periodic solutions. In the application, the tangential frequencies are ωb(ρ)=2jb−2+d+ρb and the normal blocks are Ajj=(2j−2+d)I on the Hermite eigenspaces, exactly as set up by the model (see equations (2.25)–(2.26) and the tangential/normal split in (2.24) of the paper ). The abstract theorem yields a Cantor set of parameters with meas(D\D*) ≤ ε^{1/α}, where α=32(1+(d−1)/β)(1+(4d−2)/β) in this application (Theorem 2.1 gives the general α and Theorem 2.2 specializes it to d* = d−1; see (2.14) and (2.31) ). The Hessian of the nonlinearity has the required block-decay with β∈(0,1/8) by the Hermite matrix-element estimate (Lemma 8.7) and leads to ∇^2_ζ f ∈ M_{s,β} with |∇^2_ζ f|_{s,β}≲ε, while ∇_ζ f is bounded—precisely the structure exploited by the paper’s KAM scheme (see (2.28)–(2.30) and Appendix Lemma 8.7 ). The model’s outline follows the same KAM setup and uses the same frequency map, block structure, and measure exponent α; differences are not substantive (e.g., the model describes small-divisor sets in a Diophantine |⟨ℓ,ω⟩|<γ⟨ℓ⟩^{-τ} style, whereas the paper formulates them via k and a scale-dependent κ, but both lead to the same ε^{1/α} measure estimate; see the paper’s measure section and resonant sets in Section 7 ).