1-d Quantum harmonic oscillator with time quasi-periodic quadratic perturbation: Reducibility and growth of Sobolev norms
Section snippets
Introduction and main results
Consider the one-dimensional Schrödinger equation where, we assume that
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the frequencies , , satisfy the Diophantine condition (denoted by for , ):
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the parameter , an interval , and satisfies for some ,
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is the one-dimensional quantum harmonic oscillator, i.e.
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is a quadratic form of :
Classical Hamiltonian and quantum Hamiltonian
To give some preliminary knowledge, let us recall the definition of Weyl quantization, which relates the classical and quantum mechanics, and its properties. The conclusions listed in this section can also be found in [7].
The Weyl quantization is the operator for any symbol , with , where is the Weyl operator of f: In particular, if f is a polynomial of degree at most 2 in , then is a polynomial of degree at
An abstract theorem on reducibility
Consider the 1-d time-dependent equation where is a linear differential operator, , , and the symbol is a quadratic form of with coefficients analytically depending on . More precisely, we assume that with coefficients .
Through Weyl quantization, the reducibility for the time-dependent PDE can be related to the reducibility for the -linear system :
Conjugation between classical hamiltonians
Given two quadratic classical Hamiltonians which can be presented as with and . The corresponding equations of motion are given by which are the linear systems :
Proposition 4 If the linear system is conjugated to by a time quasi-periodic -transformation, i.e.,
Proof of Theorem 1 and 2
In view of Theorem 6, to show the reducibility of Eq. (1), it is sufficient to show the reducibility of the corresponding -linear system.
For , the symbol of the quantum Hamiltonian (1) is which corresponds the quasi-periodic linear system where, for every , with , , sufficiently
Proof of Theorem 3 – 5
In this section, we show that the measure of the subset is positive for the equations (3) – (5), which implies the growths of Sobolev norm.
Acknowledgements
The authors would like to thank Prof. D. Bambusi and Prof. J. You for their interests and fruitful discussions which helped to improve the proof. They also appreciate the anonymous referees for helpful remarks and suggestions in modifying this manuscript. Z. Zhao would like to thank the support of Visiting Scholars of Shanghai Jiaotong University (SJTU) and Visiting Scholars of Chern Institute of Mathematics (CIM) during his visits.
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- 1
Z. Liang was partially supported by NSFC grant (11371097, 11571249) and Natural Science Foundation of Shanghai (19ZR1402400).
- 2
The work of Z. Zhao was partially supported by the French government through the National Research Agency (ANR) grant ANR-15-CE40-0001-03 for the project BeKAM and through the UCA JEDI Investments in the Future project managed by ANR with the reference number ANR-15-IDEX-01.
- 3
Q. Zhou was partially supported by support by NSFC grant (11671192, 11771077), the Science Fund for Distinguished Young Scholars of Tianjin (No. 19JCJQJC61300) and Nankai Zhide Foundation.