Abstract
In this paper, we establish the reducibility of a class of linear coupled quantum harmonic oscillator systems under time quasi-periodic, non-Hamiltonian, reversible perturbations. This essentially means that for most values of the frequency vector, these systems can be reduced to autonomous reversible systems with constant coefficients with respect to time. Our proof relies on an application of Kolmogorov–Arnold–Moser (KAM) theory for infinite dimensional reversible systems.
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Acknowledgements
We would like to extend our sincere gratitude to the anonymous reviewers for their valuable comments and suggestions, which significantly improved the quality of this paper. This research was supported by the National Natural Science Foundation of China(NSFC)(Grant No.11901291) and the Natural Science Foundation of Jiangsu Province, China(Grant No.BK20190395). Y. Sun was also supported by MIIT Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (Grant No.202307).
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Appendix
Appendix
In this section, we present a KAM theorem that has been proven in [24].
Consider a family of \(S-\)reversible normal form vector fields of the form
where \(\zeta \in \Pi ,\) \(\omega _b, \tilde{\omega }_b, \Omega _j,\,\tilde{\Omega }_j,\,A_j,\,\tilde{A}_j\in \mathbb {R}.\) For each \(\zeta \in \Pi \), the motion equation governed by the vector field \(X^0\) is
Obviously, \(\{(\theta +\omega t, \varphi +\tilde{\omega } t, 0,0,0,0,0,0,):t\in \mathbb {R}\}\) forms an invariant torus. Note that since in general \(\left( \begin{array}{cc} \Omega _j &{} A_j \\ \tilde{A}_j &{} \tilde{\Omega }_j \\ \end{array} \right) \) is not Hermitian, thus such invariant torus may not be elliptic.
Consider now the perturbed \(S-\)reversible vector field
We will prove that, for typical (in the sense of Lebesgue measure) \(\zeta \in \Pi \), the vector fields (4.15) still admit invariant tori for sufficiently small \(S-\)reversible P. For this purpose, we need the following assumptions.
In the sequel, for convenience, we also use the notations
Assumption 4.2
(Non-degeneracy) The map \(\zeta \mapsto \widehat{\omega }(\zeta )=(\omega (\zeta ), \tilde{\omega }(\zeta ))\) is a \(C^4_W\) diffeomorphism between \(\Pi \) and its image. Moreover there exist constants \(E_1\) and \(E_2\) such that \( |\widehat{\omega }|^{\mathcal {W}}_{\Pi } \le E_1,\) and \( |\widehat{\omega }^{-1}|^{\mathcal {W}}_{\widehat{\omega }(\Pi )} \le E_2.\)
Here we define
with
Assumption 4.3
(Asymptotics of normal frequencies) There exists constant \(\beta >0\) such that
Moreover we assume the functions
are uniformly \(C^4_W\) smooth on \(\Pi \) for all \(j\ge 1.\) And there there is a constant L such that for all \(1\le \ell \le 4,\)
Here we define
Assumption 2 implies that for \(i\ne j,\)
Furthermore set \(\Omega _0=0, \widetilde{\Omega }_0=0,\) there exist the constants \(M_0>0,\,m_0>0\) such that for all \(i, j\ge 0\) and uniformly on \(\Pi ,\)
Denote the matrix
Assumption 4.4
(Non-resonance conditions) There exist \(\alpha , \tau >0, \) such that uniformly on \(\Pi ,\)
where \(I_b\) is \(b\times b\) identity matrix. \(\det (\cdot )\), \(\otimes \) and \((\cdot )^T\) denotes the determinant, the tensor product and the transpose of matrices, respectively.
Assumption 4.5
(Regularity) Suppose there exist \(s,r>0\) such that the reversible perturbation P defines a map
Moreover, assume that \(P(\cdot ,\zeta )\) is real analytic on D(s, r) for each \(\zeta \in \Pi ,\) and \(P(\chi ,\cdot )\) is \(C^4_W-\)smooth on \(\Pi \) for each \(\chi \in D(s,r).\)
Assumption 4.6
(Decay) \(P\in \Gamma ^\beta _{D(s,r)\times \Pi } \) for some \(\beta >0\).
Now we state KAM theorem.
Theorem 4.7
([24]) Suppose \(X^0=N+\mathcal {A}\) is a family of \(S-\)reversible vector field of the form (4.13) on the phase space \(\mathscr {P}_{p}\) depending on parameters \(\zeta \in \Pi \) so that Assumptions 1–3 are satisfied. Then there exist \(0< \gamma <1\) such that for every \(S-\)reversible perturbation \(X=X^0+P\) of \(X^0\) which satisfies Assumptions 4 and 5 and the smallness condition
the following holds. There exist
-
(i)
a Cantor subset \(\Pi _\alpha \subset \Pi \) with Lebesgue measure \(Meas\left( \Pi \backslash \Pi _\alpha \right) =O\left( \alpha ^\kappa \right) \) as \(\alpha \rightarrow 0\) where \(\kappa =\min \{\frac{1}{8},\frac{\beta }{16}\};\)
-
(ii)
a \(C^4_W-\)smooth family of real analytic, \(S-\)invariant coordinate transformations
$$\begin{aligned} \Phi :D(s/2,r/2)\times \Pi _\alpha \rightarrow D(s,r); \end{aligned}$$ -
(iii)
a \(C^4_W-\)smooth family of new normal forms \(N^{\infty }+\mathcal {A}^{\infty },\) where
$$\begin{aligned} N^{\infty }=\omega ^{\infty } \frac{\partial }{\partial \theta }+\tilde{\omega }^{\infty } \frac{\partial }{\partial \varphi }+\sum _{\varrho = \pm } \varrho \textrm{i}\left( \Omega ^{\infty }(\zeta ) z^{\varrho } \frac{\partial }{\partial z^{\varrho }}+\widetilde{\Omega }^{\infty }(\zeta ) w^{\varrho } \frac{\partial }{\partial w^{\varrho }}\right) {,}\nonumber \\ \end{aligned}$$(4.20)$$\begin{aligned} \mathcal {A}^{\infty }=\sum _{\varrho = \pm } \varrho \textrm{i}\left( A^{\infty }(\zeta ) w^{\varrho } \frac{\partial }{\partial z^{\varrho }}+\tilde{A}^{\infty }(\zeta ) z^{\varrho } \frac{\partial }{\partial w^{\varrho }}\right) \nonumber \\ \end{aligned}$$(4.21)
defined on \(D(s/2,r/2)\times \Pi _\alpha ,\) such that \(\Phi ^*X=N^{\infty }+\mathcal {A}^{\infty }+P^{\infty },\) where each component \(P^{\infty ,(\textsf {v})},\) \(\textsf {v}\in \mathscr {V}\) (recall (2.5)) of \(P^{\infty }\) is real analytic on D(s/2, r/2) and of degree \(\ge 2\) at \(\mathcal {T}^{n+m}_0.\) Namely, the Taylor expansion of \(P^{\infty ,(\textsf {v})}\) only contains monomials \(I^lJ^{\tilde{l}} z^{\alpha }\bar{z}^{\beta }w^{\tilde{\alpha }}\bar{w}^{\tilde{\beta }}\) with \(|l|+|\tilde{l}|+|\alpha +\tilde{\alpha }|+|\beta +\tilde{\beta }|\ge 2.\) Moreover each \(S-\)invariant coordinate transformation is close to the identity
the new frequencies are close to the unperturbed ones
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Lou, Z., Sun, Y. & Wu, Y. Reducibility of the Linear Quantum Harmonic Oscillators Under Quasi-periodic Reversible Perturbation. Qual. Theory Dyn. Syst. 23, 224 (2024). https://doi.org/10.1007/s12346-024-01067-z
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DOI: https://doi.org/10.1007/s12346-024-01067-z