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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Decay estimates for higher-order elliptic operators
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by Hongliang Feng, Avy Soffer, Zhao Wu and Xiaohua Yao PDF
Trans. Amer. Math. Soc. 373 (2020), 2805-2859 Request permission

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
  • 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)
  • © Copyright 2020 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 373 (2020), 2805-2859
  • MSC (2010): Primary 34E05, 47F05, 81U30
  • DOI: https://doi.org/10.1090/tran/8010
  • MathSciNet review: 4069234