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Spectral inequality for Schr\"odinger equations with power growth potentials

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arxiv 2301.12338 v1 pith:RO5LL5NA submitted 2023-01-29 math.AP

Jiuyi Zhu , Jinping Zhuge This is my paper · ORCID
classification math.AP
keywords potentialsgrowthinequalityspectralequationsodingerpowerschr
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We prove a spectral inequality for Schr\"odinger equations with power growth potentials, which particularly confirms a conjecture in \cite{DSV}. This spectral inequality depends on the decaying density of the sensor sets, and the growth rate of potentials. The proof relies on three-ball inequalities derived from modified versions of quantitative global and local Carleman estimates that take advantage of the gradient information of the potentials.

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  1. Observability inequality, log-type Hausdorff content and heat equations

    math.AP 2024-11 conditional novelty 7.0 of 10

    Log-type Hausdorff contents, chosen so that their inverse matches the heat kernel, give sharp sufficient conditions and a critical-scale counterexample for heat-equation observability.

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