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Unique Continuation Properties for One Dimensional Higher Order Schrödinger Equations

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Abstract

We study two types of unique continuation properties for the higher order Schrödinger equation with potential The first one says if u has certain exponential decay at two times, then \(u\equiv 0\), and this result is sharp by constructing critical non-trivial solutions. The second one says if \(u\equiv 0\) in an arbitrary half-space of \(\mathbb {R}^{1+n}\), then \(u\equiv 0\) identically. The uniqueness theorems are given when \(n=1\), but we also prove partial results when \(n\in \mathbb {N}_+\) for their own interests. Possibility or obstacles to proving these unique continuation properties in higher spatial dimensions are also discussed.

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Acknowledgements

T. Huang is supported by the National Natural Science Foundation of China No. 12101621, the China Postdoctoral Science Foundation No. 2020M672929, and the Guangdong Basic and Applied Basic Research Foundation No. 2020A1515111048. S. Huang is supported by the National Natural Science Foundation of China Nos. 11801188, 11971188 and 12171178. Q. Zheng is supported by the National Natural Science Foundation of China Nos. 11801188 and 12171178. The Authors would like to thank the anonymous referee for carefully reading the manuscript and providing valuable comments that helped improve the exposition.

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Authors and Affiliations

  1. School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, 519082, Guangdong, China

    Tianxiao Huang

  2. School of Mathematics and Statistics, Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, 430074, Hubei, People’s Republic of China

    Shanlin Huang

  3. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, Hubei, People’s Republic of China

    Quan Zheng

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  1. Tianxiao Huang
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Huang, T., Huang, S. & Zheng, Q. Unique Continuation Properties for One Dimensional Higher Order Schrödinger Equations. J Geom Anal 32, 167 (2022). https://doi.org/10.1007/s12220-022-00906-2

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