环面上薛定谔方程的半经典测度
Semiclassical measures for the Schrödinger equation on the torus
摘要：
本文分析了环面上线性薛定谔方程解的半经典测度的结构。我们证明了这样的测度在每个不变拉格朗日环面上的分解相对于勒贝格测度是绝对连续的。我们得到的Radon-Nikodym衍生物的序列的初始数据的表达式，并表明它满足一个明确的传播规律。作为结果，我们还证明了一个可观性不等式，说的$L^2$ -范数的任何开子集的环面上的解决方案控制的全部$L^2$ -范数。
英文摘要：
In this article, the structure of semiclassical measures for solutions to the linear Schrödinger equation on the torus is analysed. We show that the disintegration of such a measure on every invariant lagrangian torus is absolutely continuous with respect to the Lebesgue measure. We obtain an expression of the Radon-Nikodym derivative in terms of the sequence of initial data and show that it satisfies an explicit propagation law. As a consequence, we also prove an observability inequality, saying that the $L^2$-norm of a solution on any open subset of the torus controls the full $L^2$-norm.
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