Reducibility of quantum harmonic oscillator on $$\mathbb {R}^d$$ perturbed by a quasi: periodic potential with logarithmic decay

Reducibility of 1-D Quantum Harmonic Oscillator with New Unbounded Oscillatory Perturbations

“…Eliasson-Kuksin [13] initiated to prove the reducibility for PDEs in high dimension. See [22] and [26] for higher-dimensional QHO with bounded potential. The first reducibility result for n-D QHO was proved in [7] by Bambusi-Grébert-Maspero-Robert.…”

“…However, when α ∈ (0, 1 2 ), we can't insure that M α ⊂ L(ℓ 2 0 , ℓ 2 s ) for any s ∈ R. This means that P x makes no sense when the perturbation operator P ∈ M α and x ∈ ℓ 2 0 . Fortunately, from Lemma 2.1 in [22] or Lemma 2.2 in [26] one can show M α ⊂ L(ℓ 2 1 , ℓ 2 −1 ) and thus the reducibility in H 1 can be built up in Theorem 1.1 instead of L 2 .…”

Reducibility of 1D quantum harmonic oscillator with decaying conditions on the derivative of perturbation potentials

“…From (1.7), one gets λ i = ν i + C (1 + ln i) −2δ 2 and then if choose small divisor conditions carefully, one can set up the reducibility result as above. We remark that the result in [45] has recently been generalized to any higher dimensions in [33]. Now we will explain how we can estimate the set (1.6).…”

“…Reducibility for PDEs in high dimension was initiated by Eliasson-Kuksin [19]. We can refer to [27,32,33] for higher-dimensional QHO with bounded potential. The reducibility result for nD QHO with polynomial perturbations was first set up in [6].…”