Uncertainty Principle and Geometric Condition for the Observability of Schr\"{o}dinger Equations

Longben Wei; Zhiwen Duan; Hui Xu

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arXiv · 2307.09592

Uncertainty Principle and Geometric Condition for the Observability of Schr\"{o}dinger Equations

Longben Wei·Zhiwen Duan·Hui Xu

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Abstract

We provide necessary and sufficient geometric conditions for the exact observability of the Schr\"odinger equation with inverse-square potentials on the half-line. These conditions are derived from a Logvinenko-Sereda type theorem for generalized Fourier transform. Specifically, the generalized Fourier transform associated with the Schr\"odinger operator with inverse-square potentials on the half-line is the well-known Hankel transform. We present a necessary and sufficient condition for a subset $\Omega$, such that a function whose Hankel transform is supported in a given interval can be bounded, in the $L^2$-norm, from above by its restriction to $\Omega$, with a constant independent of the position of the interval.

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Longben Wei, Zhiwen Duan, Hui Xu. 2023-07-13. Uncertainty Principle and Geometric Condition for the Observability of Schr\"{o}dinger Equations. https://arxiv.org/abs/2307.09592

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