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Unique continuation for the Schrödinger equation with gradient term


Authors: Youngwoo Koh and Ihyeok Seo
Journal: Proc. Amer. Math. Soc. 146 (2018), 2555-2562
MSC (2010): Primary 35B60, 35B45; Secondary 35Q40
DOI: https://doi.org/10.1090/proc/13942
Published electronically: February 1, 2018
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Abstract: We obtain a unique continuation result for the differential inequality $ \vert (i\partial _t +\Delta )u \vert \leq \vert Vu\vert + \vert W\cdot \nabla u \vert$ by establishing $ L^2$ Carleman estimates. Here, $ V$ is a scalar function and $ W$ is a vector function, which may be time-dependent or time-independent. As a consequence, we give a similar result for the magnetic Schrödinger equation.


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Additional Information

Youngwoo Koh
Affiliation: Department of Mathematics Education, Kongju National University, Kongju 32588, Republic of Korea
Email: ywkoh@kongju.ac.kr

Ihyeok Seo
Affiliation: Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea
Email: ihseo@skku.edu

DOI: https://doi.org/10.1090/proc/13942
Keywords: Unique continuation, Carleman estimates, Schr\"odinger equation.
Received by editor(s): June 9, 2017
Received by editor(s) in revised form: August 28, 2017, and September 1, 2017
Published electronically: February 1, 2018
Additional Notes: The first author was supported by NRF Grant 2016R1D1A1B03932049 (Republic of Korea). The second author was supported by the NRF grant funded by the Korea government(MSIP) (No. 2017R1C1B5017496).
Communicated by: Joachim Krieger
Article copyright: © Copyright 2018 American Mathematical Society

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