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From resolvent estimates to unique continuation for the Schrödinger equation


Author: Ihyeok Seo
Journal: Trans. Amer. Math. Soc. 368 (2016), 8755-8784
MSC (2010): Primary 47A10, 35B60; Secondary 35Q40
DOI: https://doi.org/10.1090/tran/6635
Published electronically: January 13, 2016
MathSciNet review: 3551588
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Abstract: In this paper we develop an abstract method to handle the problem of unique continuation for the Schrödinger equation $ (i\partial _t+\Delta )u=V(x)u$. In general the problem is to find a class of potentials $ V$ which allows the unique continuation. The key point of our work is to make a direct link between the problem and the weighted $ L^2$ resolvent estimates $ \Vert(-\Delta -z)^{-1}f\Vert _{L^2(\vert V\vert)}\leq C\Vert f\Vert _{L^2(\vert V\vert^{-1})}$. We carry it out in an abstract way, and thereby we do not need to deal with each of the potential classes. To do so, we will make use of the limiting absorption principle and Kato $ H$-smoothing theorem in spectral theory, and employ some tools from harmonic analysis. Once the resolvent estimate is set up for a potential class, from our abstract theory the unique continuation would follow from the same potential class. Also, it turns out that there can be no dented surface on the boundary of the maximal open zero set of the solution $ u$. In this regard, another main issue for us is to know which class of potentials allows the resolvent estimate. We establish such a new class which contains previously known ones, and we will also apply it to the problem of well-posedness for the equation.


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Additional Information

Ihyeok Seo
Affiliation: Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea
Email: ihseo@skku.edu

DOI: https://doi.org/10.1090/tran/6635
Keywords: Resolvent estimates, unique continuation, Schr\"odinger equations
Received by editor(s): September 3, 2014
Received by editor(s) in revised form: November 2, 2015, and November 15, 2014
Published electronically: January 13, 2016
Article copyright: © Copyright 2016 American Mathematical Society