On weakly turbulent solutions to the perturbed linear harmonic oscillator

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2023-10-01 DOI:10.1353/ajm.2023.a907703

Erwan Faou, Pierre Raphaël

引用次数: 16

Abstract

abstract: We introduce specific solutions to the linear harmonic oscillator, named bubbles. They form resonant families of invariant tori of the linear dynamics, with arbitrarily large Sobolev norms. We use these modulated bubbles of energy to construct a class of potentials which are real, smooth, time dependent and uniformly decaying to zero with respect to time, such that the corresponding perturbed quantum harmonic oscillator admits solutions which exhibit a logarithmic growth of Sobolev norms. The resonance mechanism is explicit in space variables and produces highly oscillatory solutions. We then give several recipes to construct similar examples using more specific tools based on the continuous resonant (CR) equation in dimension two.

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摄动线性谐振子的弱湍流解

我们介绍了线性谐振子气泡的具体解。它们形成线性动力学的不变环面共振族，具有任意大的索博列夫范数。我们使用这些调制的能量泡来构造一类真实的、光滑的、与时间相关的、随时间均匀衰减到零的势，使得相应的摄动量子谐振子允许具有索博列夫范数对数增长的解。共振机制在空间变量中是显式的，并产生高振荡解。然后，我们给出了几个方法，使用基于二维连续谐振(CR)方程的更具体的工具来构建类似的例子。

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

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.