论文标题
在扰动线性谐波振荡器的弱湍流解决方案上
On weakly turbulent solutions to the perturbed linear Harmonic oscillator
论文作者
论文摘要
我们将特定的解决方案引入线性谐波振荡器，名为Bubbles。它们形成了线性动力学不变的托里的共振家族，并具有任意较大的Sobolev规范。我们使用这些调制的能量气泡来构建一类电位，这些电势相对于时间而言，这些电势相对于时间而言，均匀腐烂至零，以使相应的扰动量子谐波振荡器允许溶液显示出Sobolev规范的对数生长的溶液。共振机制在空间变量中是明确的，并产生高度振荡的解决方案。然后，我们使用基于二维式谐振（CR）方程的更具体的工具来构建类似的示例，以构建类似的示例。
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.
