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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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Dispersive estimates for Schrödinger operators in dimension two with obstructions at zero energy
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by M. Burak Erdoğan and William R. Green PDF
Trans. Amer. Math. Soc. 365 (2013), 6403-6440 Request permission

Abstract:

We investigate $L^1(\mathbb {R}^2)\to L^\infty (\mathbb {R}^2)$ dispersive estimates for the Schrödinger operator $H=-\Delta +V$ when there are obstructions, resonances or an eigenvalue, at zero energy. In particular, we show that the existence of an s-wave resonance at zero energy does not destroy the $t^{-1}$ decay rate. We also show that if there is a p-wave resonance or an eigenvalue at zero energy, then there is a time dependent operator $F_t$ satisfying $\|F_t\|_{L^1\to L^\infty } \lesssim 1$ such that \[ \|e^{itH}P_{ac}-F_t\|_{L^1\to L^\infty } \lesssim |t|^{-1}, \text { for } |t|>1.\] We also establish a weighted dispersive estimate with $t^{-1}$ decay rate in the case when there is an eigenvalue at zero energy but no resonances.
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Additional Information
  • M. Burak Erdoğan
  • Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
  • Email: berdogan@math.uiuc.edu
  • William R. Green
  • Affiliation: Department of Mathematics and Computer Science, Eastern Illinois University, Charleston, Illinois 61920
  • Address at time of publication: Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute, Indiana 47803
  • MR Author ID: 906481
  • ORCID: 0000-0001-9399-8380
  • Email: wrgreen2@eiu.edu, green@rose-hulman.edu
  • Received by editor(s): January 11, 2012
  • Received by editor(s) in revised form: April 19, 2012
  • Published electronically: April 18, 2013
  • Additional Notes: The first author was partially supported by NSF grant DMS-0900865.
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 365 (2013), 6403-6440
  • MSC (2010): Primary 35J10
  • DOI: https://doi.org/10.1090/S0002-9947-2013-05861-8
  • MathSciNet review: 3105757