We provide necessary and sufficient geometric conditions for the exact controllability of the one-dimensional fractional free and fractional harmonic Schrödinger equations. The necessary and sufficient condition for the exact controllability of fractional free Schrödinger equations is derived from the Logvinenko–Sereda theorem and its quantitative version established by Kovrijkine, whereas the one for the exact controllability of fractional harmonic Schrödinger equations is deduced from an infinite dimensional version of the Hautus test for Hermite functions and the Plancherel–Rotach formula.

中文翻译：

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分数阶自由和谐波Schrödinger方程的精确可控制性的几何条件

我们为一维分数阶自由和分数次谐波Schrödinger方程的精确可控制性提供了必要和充分的几何条件。Logvinenko-Sereda定理及其由Kovrijkine建立的定量形式推导了分数阶自由Schrödinger方程的精确可控制性的充要条件，而分数次谐波Schrödinger方程的精确可控制性的一个条件是从无穷维形式的用于Hermite函数和Plancherel-Rotach公式的Hautus检验。