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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Dispersive estimates for Schrödinger operators in dimension two with obstructions at zero energy


Authors: M. Burak Erdoğan and William R. Green
Journal: Trans. Amer. Math. Soc. 365 (2013), 6403-6440
MSC (2010): Primary 35J10
Published electronically: April 18, 2013
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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 $ \Vert F_t\Vert _{L^1\to L^\infty } \lesssim 1$ such that

$\displaystyle \Vert e^{itH}P_{ac}-F_t\Vert _{L^1\to L^\infty } \lesssim \vert t\vert^{-1},\,\,\,\,\,$$\displaystyle \text { for } \vert t\vert>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
Email: wrgreen2@eiu.edu, green@rose-hulman.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05861-8
PII: S 0002-9947(2013)05861-8
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.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.