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Transactions of the American Mathematical Society

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Decay estimates for higher-order elliptic operators


Authors: Hongliang Feng, Avy Soffer, Zhao Wu and Xiaohua Yao
Journal: Trans. Amer. Math. Soc. 373 (2020), 2805-2859
MSC (2010): Primary 34E05, 47F05, 81U30
DOI: https://doi.org/10.1090/tran/8010
Published electronically: January 28, 2020
MathSciNet review: 4069234
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Abstract: This paper is mainly devoted to the study of time decay estimates of the higher-order Schrödinger-type operator $H=(-\Delta )^{m}+V(x)$ in $\mathbf {R}^{n}$ for $n>2m$ and $m\in \mathbf {N}$. For certain decay potentials $V(x)$, we first derive the asymptotic expansions of resolvent $R_{V}(z)$ near zero threshold with the presence of zero resonance or zero eigenvalue, as well as identify the resonance space for each kind of zero resonance which displays different effects on time decay rate. Then we establish Kato-Jensen-type estimates and local decay estimates for higher-order Schrödinger propagator $e^{-itH}$ in the presence of zero resonance or zero eigenvalue. As a consequence, the endpoint Strichartz estimate and $L^{p}$-decay estimates can also be obtained. Finally, by a virial argument, a criterion on the absence of positive embedding eigenvalues is given for $(-\Delta )^{m}+V(x)$ with a repulsive potential.


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Additional Information

Hongliang Feng
Affiliation: School of Mathematics, Sun Yat-sen University, Guangzhou, 510275, People’s Republic of China
MR Author ID: 1241378
Email: fenghongliang@aliyun.com

Avy Soffer
Affiliation: School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, People’s Republic of China
Email: soffer@math.rutgers.edu

Zhao Wu
Affiliation: School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, People’s Republic of China
Email: wuzhao218@yahoo.com

Xiaohua Yao
Affiliation: Department of Mathematics and Hubei Province Key Laboratory of Mathematical Physics, Central China Normal University, Wuhan, 430079, People’s Republic of China
MR Author ID: 680187
Email: yaoxiaohua@mail.ccnu.edu.cn

Keywords: Higher-order Schrödinger-type operator, lower energy asymptotic expansion, zero resonance, Kato-Jensen decay estimates, Strichartz estimates, absence of positive eiganvalue
Received by editor(s): May 14, 2019
Received by editor(s) in revised form: September 10, 2019
Published electronically: January 28, 2020
Additional Notes: Xiaohua Yao is the corresponding author
The first author was supported by the China Postdoctoral Science Fundation, Grant No. 2019M653135
The second author was partially supported by NSFC grant No. 11671163 and NSF grant DMS-1600749
The fourth author was partially supported by NSFC (No. 11771165) and the program for Changjiang Scholars and Innovative Research Team in University (IRT13066)
Article copyright: © Copyright 2020 American Mathematical Society