abstract:

We develop a new abstract derivation of the observability inequalities at two points in time for Schr\"{o}dinger type equations. Our approach consists of two steps. In the first step we prove a Nazarov type uncertainty principle associated with a non-negative self-adjoint operator $H$ on $L^2(\Bbb{R}^n)$. In the second step we use results on asymptotic behavior of $e^{-itH}$, in particular, minimal velocity estimates introduced by Sigal and Soffer. Such observability inequalities are closely related to unique continuation problems as well as controllability for the Schr\"{o}dinger equation.

中文翻译：

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薛定谔方程的不确定性原理、最小逃逸速度和可观测性不等式

摘要：

我们开发了 Schr\"{o}dinger 型方程在两个时间点的可观察性不等式的新抽象推导。我们的方法包括两个步骤。第一步，我们证明了与非负相关的 Nazarov 型不确定性原理$L^2(\Bbb{R}^n)$ 上的自伴随算子 $H$。在第二步中，我们使用 $e^{-itH}$ 的渐近行为的结果，特别是引入了最小速度估计由 Sigal 和 Soffer 撰写。这种可观察性不等式与独特的延拓问题以及 Schr\"{o}dinger 方程的可控性密切相关。