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Transactions of the American Mathematical Society

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Spatial asymptotics of Green's function for elliptic operators and applications: a.c. spectral type, wave operators for wave equation


Author: Sergey A. Denisov
Journal: Trans. Amer. Math. Soc. 371 (2019), 8907-8970
MSC (2010): Primary 35L05, 81Q10; Secondary 35Q40, 35P25
DOI: https://doi.org/10.1090/tran/7800
Published electronically: March 11, 2019
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Abstract: In the three-dimensional case, we consider a Schrödinger operator and an elliptic operator in the divergence form. For slowly decaying oscillating potentials, we establish spatial asymptotics of the Green's function. The main term in this asymptotics involves an $ L^2(\mathbb{S}^2)$-valued analytic function whose behavior is studied away from the spectrum. This analysis is used to prove that the absolutely continuous spectrum of both operators fills $ \mathbb{R}^+$. We also apply our technique to establish the existence of the wave operators for a wave equation under optimal conditions for decay of the potential.


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Additional Information

Sergey A. Denisov
Affiliation: Department of Mathematics, University of Wisconsin–Madison, 480 Lincoln Drive, Madison, Wisconsin 53706; and Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya ploshad 4, 125047 Moscow, Russia
Email: denissov@wisc.edu

DOI: https://doi.org/10.1090/tran/7800
Received by editor(s): December 19, 2018
Received by editor(s) in revised form: January 7, 2019
Published electronically: March 11, 2019
Additional Notes: The work done in the last section of the paper was supported by a grant of the Russian Science Foundation (project RScF-14-21-00025), and research conducted in the rest of the paper was supported by grants NSF-DMS-1464479 and NSF DMS-1764245 and the Van Vleck Professorship Research Award. The author gratefully acknowledges the hospitality of IHES, where part of this work was done.
Article copyright: © Copyright 2019 American Mathematical Society