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Reducibility of the Linear Quantum Harmonic Oscillators Under Quasi-periodic Reversible Perturbation

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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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Authors and Affiliations

  1. School of Mathematics, Nanjing University of Aeronautics and Astronautics, MIIT Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles, Nanjing, 211106, China

    Zhaowei Lou & Yingnan Sun

  2. School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China

    Youchao Wu

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  1. Zhaowei Lou
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  2. Yingnan Sun
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YW wrote the main manuscript text, ZL and YS edited the manuscript. All authors reviewed the manuscript.

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Correspondence to Yingnan Sun.

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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

$$\begin{aligned}{} & {} \begin{aligned} X^0(\zeta )=&N(\zeta )+\mathcal {A}(\zeta ), \\ N=&\sum _{b=1}^n \omega _b(\zeta ) \frac{\partial }{\partial \theta _b}+\sum _{b=1}^m \tilde{\omega }_b(\zeta ) \frac{\partial }{\partial \varphi _b} \\&+\sum _{\varrho = \pm }\left( \sum _{j \ge 1} \varrho i \Omega _j(\zeta ) z_j^{\varrho } \frac{\partial }{\partial z_j^\rho }+\sum _{j \ge 1} \varrho i \widetilde{\Omega }_j(\zeta ) w_j^{\varrho } \frac{\partial }{\partial w_j^{\varrho }}\right) {,} \end{aligned} \end{aligned}$$
(4.13)
$$\begin{aligned} \mathcal {A}= & {} \sum _{\varrho = \pm } \sum _{j \ge 1}\left( \varrho \textrm{i} A_j(\zeta ) w_j^\rho \frac{\partial }{\partial z_j^\rho }+\varrho \textrm{i} \tilde{A}_j(\zeta ) z_j^{\varrho } \frac{\partial }{\partial w_j^{\varrho }}\right) \end{aligned}$$
(4.14)

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

$$\begin{aligned} \left\{ \begin{array}{l} \dot{\theta }=\omega , \dot{\varphi }=\tilde{\omega },\\ \dot{I}=0, \dot{J}=0,\\ \left( \begin{array}{c} \dot{z}_j^\sigma \\ \dot{w}_j^\sigma \end{array}\right) =\sigma \textrm{i}\left( \begin{array}{cc} \Omega _j &{} A_j \\ \tilde{A}_j &{} \tilde{\Omega }_j \end{array}\right) \left( \begin{array}{c} z_j^\sigma \\ w_j^\sigma \end{array}\right) , \quad \sigma = \pm , j \ge 1. \end{array}\right. \end{aligned}$$

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

$$\begin{aligned} X(\chi ;\zeta )=X^0(\zeta )+P(\chi ;\zeta )=N(\zeta )+\mathcal {A}(\zeta )+P(\chi ;\zeta ). \end{aligned}$$
(4.15)

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

$$\begin{aligned} \hat{\theta }=(\theta , \varphi )\in \mathbb {T}^{n}\times \mathbb {T}^{m},\,\,\widehat{\omega }=(\omega , \tilde{\omega })\in \mathbb {R}^{n}\times \mathbb {R}^{m},\,\,\hat{k}=(k, \tilde{k})\in \mathbb {Z}^{n}\times \mathbb {Z}^{m}, \\ \langle \hat{k}, \hat{\theta }\rangle =\langle k, \theta \rangle +\langle \tilde{k}, \varphi \rangle ,\,\,\langle \hat{k}, \widehat{\omega }\rangle =\langle k, \omega \rangle +\langle \tilde{k}, \tilde{\omega }\rangle ,\,\, |\hat{k}|=|k|+|\tilde{k}|. \end{aligned}$$

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

$$\begin{aligned} |\widehat{\omega }|^{\mathcal {W}}_{\Pi } = |\widehat{\omega }|_{\Pi }+\sum ^4_{\ell =1}|\widehat{\omega }|^{C^\ell _W}_{\Pi } \end{aligned}$$

with

$$\begin{aligned} |\widehat{\omega }|_{\Pi } =\sup \limits _{ \zeta \in \Pi }|\widehat{\omega }(\zeta )|\,\,\,\hbox {and}\,\,\,|\widehat{\omega }|^{C^\ell _W}_{\Pi } =\sup \limits _{ \zeta \in \Pi }|\partial ^\ell _{\zeta } \widehat{\omega }(\zeta )|. \end{aligned}$$

Assumption 4.3

(Asymptotics of normal frequencies) There exists constant \(\beta >0\) such that

$$\begin{aligned} \Omega _j,\tilde{\Omega }_j=2j-1+O\left( \frac{1}{j^{2\beta }}\right) . \end{aligned}$$
(4.16)

Moreover we assume the functions

$$\begin{aligned} \zeta \mapsto j^{2\beta }\Omega _j(\zeta ),\,\, \zeta \mapsto j^{2\beta }\tilde{\Omega }_j(\zeta ),\,\, \zeta \mapsto j^{2\beta }A_j(\zeta ),\,\, \zeta \mapsto j^{2\beta }\tilde{A}_j(\zeta ), \end{aligned}$$

are uniformly \(C^4_W\) smooth on \(\Pi \) for all \(j\ge 1.\) And there there is a constant L such that for all \(1\le \ell \le 4,\)

$$\begin{aligned} |\Omega |^{{C^\ell _W}}_{2\beta ,\Pi },\, \, |\tilde{\Omega }|^{{C^\ell _W}}_{2\beta ,\Pi },\,\, |A|^{{C^\ell _W}}_{2\beta ,\Pi },\, \, |\tilde{A}|^{{C^\ell _W}}_{2\beta ,\Pi }\le L. \end{aligned}$$

Here we define

$$\begin{aligned} |\Omega |^{{C^\ell _W}}_{2\beta ,\Pi } =\sup \limits _{\zeta \in \Pi }\sup \limits _{j\ge 1 }j^{2\beta }|\partial ^\ell _{\zeta } \Omega _j(\zeta )|. \end{aligned}$$

Assumption 2 implies that for \(i\ne j,\)

$$\begin{aligned} \left| \frac{\Omega _i-\Omega _j}{2(i-j)}\right| ,\left| \frac{\Omega _i-\tilde{\Omega }_j}{2(i-j)}\right| ,\left| \frac{\tilde{\Omega }_i-\Omega _j}{2(i-j)}\right| ,\left| \frac{\tilde{\Omega }_i-\tilde{\Omega }_j}{2(i-j)}\right| =1+O\left( \frac{1}{i^{2\beta }}+\frac{1}{j^{2\beta }}\right) .\nonumber \\ \end{aligned}$$
(4.17)

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 \(\Pi ,\)

$$\begin{aligned} \begin{aligned} m_0|i-j|\le&|\Omega _i-\Omega _j|\le M_0|i-j|,\\ m_0|i-j|\le&|\Omega _i-\tilde{\Omega }_j|\le M_0|i-j|, \\ m_0|i-j|\le&|\tilde{\Omega }_i-\Omega _j|\le M_0|i-j|, \\ m_0|i-j|\le&|\tilde{\Omega }_i-\tilde{\Omega }_j|\le M_0|i-j|. \end{aligned} \end{aligned}$$
(4.18)

Denote the matrix

$$\begin{aligned} M_j\,=\left( \begin{array}{cc} \Omega _j &{} A_j \\ \tilde{A}_j &{} \tilde{\Omega }_j \\ \end{array} \right) ,\,j\ge 1. \end{aligned}$$

Assumption 4.4

(Non-resonance conditions) There exist \(\alpha , \tau >0, \) such that uniformly on \(\Pi ,\)

$$\begin{aligned} \begin{aligned}&|\langle \hat{k},\widehat{\omega }\rangle |\ge \frac{\alpha }{|\hat{k}|^\tau },\,\, \hat{k}\ne 0{,}\\&|\det (\langle \hat{k},\widehat{\omega }\rangle I_2\pm M_i)|\ge \frac{\alpha i}{|\hat{k}|^\tau },\hat{k}\ne 0,\,i\ge 1, \\&|\det (\langle \hat{k},\widehat{\omega }\rangle I_{4}+ M_i\otimes I_2\pm I_2\otimes M^T_j)|\ge \frac{\alpha |i\pm j|}{|\hat{k}|^\tau },\,\,\hat{k}\ne 0, i, j\ge 1. \\ \end{aligned} \end{aligned}$$
(4.19)

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

$$\begin{aligned} P: D(s,r)\times \Pi \rightarrow \mathscr {P}_{p}. \end{aligned}$$

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

$$\begin{aligned} \varepsilon =\Vert P\Vert ^{\alpha ,\mathcal {W}}_{r,D(s,r)\times \Pi }+\langle P\rangle ^{\alpha ,\mathcal {W}}_{D(s,r)\times \Pi }\le \gamma \alpha ^2, \end{aligned}$$

the following holds. There exist

  1. (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 \(\kappa =\min \{\frac{1}{8},\frac{\beta }{16}\};\)

  2. (ii)

    a \(C^4_W-\)smooth family of real analytic, \(S-\)invariant coordinate transformations

    $$\begin{aligned} \Phi :D(s/2,r/2)\times \Pi _\alpha \rightarrow D(s,r); \end{aligned}$$
  3. (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

$$\begin{aligned} \Vert \Phi -id\Vert ^{\mathcal {W}}_{r;D(s/2,r/2)\times \Pi _\alpha },\,\,\Vert D\Phi -Id\Vert ^{\mathcal {W}}_{r;D(s/2,r/2)\times \Pi _\alpha }\le c\varepsilon ^{\frac{1}{4}}, \end{aligned}$$
(4.22)

the new frequencies are close to the unperturbed ones

$$\begin{aligned} \begin{aligned} |\hat{\omega }^{\infty }-\hat{\omega }|^{\mathcal {W}}_{\Pi _\alpha }&\le c\varepsilon ^{\frac{1}{2}},\\ \end{aligned} \end{aligned}$$
(4.23)
$$\begin{aligned} |\Omega ^{\infty }-\Omega |^{\alpha /2,\mathcal {W} }_{2\beta , \Pi _\alpha },\,\, |\tilde{\Omega }^{\infty }-\tilde{\Omega }|^{\alpha /2,\mathcal {W} }_{2\beta , \Pi _\alpha }\le c\varepsilon ^{\frac{1}{2}}, \end{aligned}$$
(4.24)
$$\begin{aligned} |A^{\infty }-A|^{\alpha /2,\mathcal {W} }_{2\beta , \Pi _\alpha },\,\, |\tilde{A}^{\infty }-\tilde{A}|^{\alpha /2,\mathcal {W} }_{2\beta , \Pi _\alpha }\le c\varepsilon ^{\frac{1}{2}}. \end{aligned}$$
(4.25)

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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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