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Global-in-time smoothing effects for Schrödinger equations with inverse-square potentials


Author: Haruya Mizutani
Journal: Proc. Amer. Math. Soc. 146 (2018), 295-307
MSC (2010): Primary 35Q41; Secondary 35B45
DOI: https://doi.org/10.1090/proc/13729
Published electronically: July 27, 2017
MathSciNet review: 3723141
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Abstract: The purpose of this note is to prove global-in-time smoothing effects for the Schrödinger equation with potentials exhibiting critical singularity. A typical example of admissible potentials is the inverse-square potential $ a\vert x\vert^{-2}$ with $ a>-(n-2)^2/4$. This particularly gives an affirmative answer to a question raised by T. A. Bui et al. (J. Differential Equations 262 (2017), 2771-2807). The proof employs a uniform resolvent estimate proved by Barceló, Vega, and Zubeldia (Adv. Math. 240 (2013), 636-671) an abstract perturbation method by Bouclet and Mizutani (preprint).


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Additional Information

Haruya Mizutani
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
Email: haruya@math.sci.osaka-u.ac.jp

DOI: https://doi.org/10.1090/proc/13729
Keywords: Smoothing estimate, Strichartz estimate, Schr\"odinger equation, inverse-square potential
Received by editor(s): December 21, 2016
Received by editor(s) in revised form: March 3, 2017
Published electronically: July 27, 2017
Additional Notes: The author was partially supported by JSPS Grant-in-Aid for Young Scientists (B) JP25800083 and by Osaka University Research Abroad Program 150S007
Communicated by: Joachim Krieger
Article copyright: © Copyright 2017 American Mathematical Society