论文标题
用二次哈密顿人的Schrödinger方程可观察性
Observability for Schrödinger equations with quadratic Hamiltonians
论文作者
论文摘要
我们考虑使用高斯波袋为相应的schrödinger方程构造时间依赖的谐波振荡器，并构建一个参数。高斯波袋的这种参数是精确且可拖延的。使用此参数，我们证明$ l^2 $和$ l^2-l^{\ infty} $可观察性估算值限制了限制类的初始数据类别。该数据包括一类紧凑的分段$ c^1 $函数，这些功能已从特征函数扩展。可以观察到该形式的大部分质量数据，其大部分质量远离$ω^c =ω$是一个连接的有限域，但是以$ω$为中心的数据必须几乎是一个高斯，才能观察到。在特定时间依赖的谐波振荡器的情况下，我们还为简单的谐波振荡器的既定原则提供了反例。
We consider time dependent harmonic oscillators and construct a parametrix to the corresponding Schrödinger equation using Gaussian wavepackets. This parametrix of Gaussian wavepackets is precise and tractable. Using this parametrix we prove $L^2$ and $L^2-L^{\infty}$ observability estimates on unbounded domains $ω$ for a restricted class of initial data. This data includes a class of compactly supported piecewise $C^1$ functions which have been extended from characteristic functions. Initial data of this form which has the bulk of its mass away from $ω^c=Ω$, a connected bounded domain, is observable, but data centered over $Ω$ must be very nearly a single Gaussian to be observable. We also give counterexamples to established principles for the simple harmonic oscillator in the case of certain time dependent harmonic oscillators.
