On the reducibility of a class of almost-periodic linear Hamiltonian systems and its application in Schrödinger equation

Muhammad Afzal; Tariq Ismaeel; Azhar Iqbal Kashif Butt; Zahid Farooq; Riaz Ahmad; Ilyas Khan; Muhammad Afzal; Tariq Ismaeel; Azhar Iqbal Kashif Butt; Zahid Farooq; Riaz Ahmad; Ilyas Khan

doi:10.3934/math.2023375

AIMS Mathematics

2023, Volume 8, Issue 3: 7471-7489. doi: 10.3934/math.2023375

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Research article

On the reducibility of a class of almost-periodic linear Hamiltonian systems and its application in Schrödinger equation

1.
Department of Mathematics, Division of Science and Technology, University of Education, Lahore 54770, Pakistan
2.
Department of Mathematics, Government College University, Lahore 54000, Pakistan
3.
Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia
4.
Department of Physics, Division of Science and Technology, University of Education, Lahore 54770, Pakistan
5.
Department of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, China
6.
Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah, P.O. Box 66, Majmaah 11952, Saudi Arabia

Received: 27 October 2022 Revised: 26 December 2022 Accepted: 02 January 2023 Published: 16 January 2023
MSC : 37K55, 70K40

In the present paper, we focus on the reducibility of an almost-periodic linear Hamiltonian system

$ \frac{dX}{dt} = J[A+\varepsilon Q(t)]X, X\in \mathbb{R}^{2d} , $

where $ J $ is an anti-symmetric symplectic matrix, $ A $ is a symmetric matrix, $ Q(t) $ is an analytic almost-periodic matrix with respect to $ t $, and $ \varepsilon $ is a parameter which is sufficiently small. Using some non-resonant and non-degeneracy conditions, rapidly convergent methods prove that, for most sufficiently small $ \varepsilon $, the Hamiltonian system is reducible to a constant coefficients Hamiltonian system through an almost-periodic symplectic transformation with similar frequencies as $ Q(t) $. At the end, an application to Schrödinger equation is given.
- reducibility,
- almost-periodic,
- KAM method,
- Hamiltonian system
Citation: Muhammad Afzal, Tariq Ismaeel, Azhar Iqbal Kashif Butt, Zahid Farooq, Riaz Ahmad, Ilyas Khan. On the reducibility of a class of almost-periodic linear Hamiltonian systems and its application in Schrödinger equation[J]. AIMS Mathematics, 2023, 8(3): 7471-7489. doi: 10.3934/math.2023375

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Abstract

In the present paper, we focus on the reducibility of an almost-periodic linear Hamiltonian system

$ \frac{dX}{dt} = J[A+\varepsilon Q(t)]X, X\in \mathbb{R}^{2d} , $

where $ J $ is an anti-symmetric symplectic matrix, $ A $ is a symmetric matrix, $ Q(t) $ is an analytic almost-periodic matrix with respect to $ t $, and $ \varepsilon $ is a parameter which is sufficiently small. Using some non-resonant and non-degeneracy conditions, rapidly convergent methods prove that, for most sufficiently small $ \varepsilon $, the Hamiltonian system is reducible to a constant coefficients Hamiltonian system through an almost-periodic symplectic transformation with similar frequencies as $ Q(t) $. At the end, an application to Schrödinger equation is given.

References

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