Quasi-periodic solutions for the nonlinear quantum harmonic oscillator

IF 2.9 3区数学 Q1 MATHEMATICS, APPLIED

Physica D: Nonlinear Phenomena Pub Date : 2025-07-26 DOI:10.1016/j.physd.2025.134846

Jianjun Liu , Caihong Qi , Guanghua Shi

引用次数: 0

Abstract

This paper is concerned with the nonlinear quantum harmonic oscillator equation

i u_{t} = - u_{x x} + x^{2} u + {| u |}^{2} u, x \in R .

It is proved that there are Cantor families of 2-dimensional elliptic invariant tori, and thus quasi-periodic solutions for the above equation. The proof is based on infinite dimensional KAM theory and partial Birkhoff normal form. Compared with previous KAM results for quantum harmonic oscillator, the novelty lies in the above equation not containing external parameters. This gives rise to the difficulty associated with complete resonance of frequencies.

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非线性量子谐振子的准周期解

本文研究非线性量子谐振子方程iut=−uxx+x2u+|u|2u,x∈R。证明了二维椭圆不变环面存在Cantor族，从而证明了上述方程的拟周期解。该证明基于无限维KAM理论和部分Birkhoff范式。与以往量子谐振子的KAM结果相比，新颖性在于上述方程不包含外部参数。这引起了与频率完全共振相关的困难。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Physica D: Nonlinear Phenomena 物理-物理：数学物理

CiteScore

7.30

自引率

7.50%

发文量

213

审稿时长

65 days

期刊介绍： Physica D (Nonlinear Phenomena) publishes research and review articles reporting on experimental and theoretical works, techniques and ideas that advance the understanding of nonlinear phenomena. Topics encompass wave motion in physical, chemical and biological systems; physical or biological phenomena governed by nonlinear field equations, including hydrodynamics and turbulence; pattern formation and cooperative phenomena; instability, bifurcations, chaos, and space-time disorder; integrable/Hamiltonian systems; asymptotic analysis and, more generally, mathematical methods for nonlinear systems.