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Some remarks on resonances in even-dimensional Euclidean scattering


Authors: T. J. Christiansen and P. D. Hislop
Journal: Trans. Amer. Math. Soc. 368 (2016), 1361-1385
MSC (2010): Primary 35P25; Secondary 81U05, 47A40
Published electronically: December 19, 2014
MathSciNet review: 3430366
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Abstract:

Black box quantum mechanical scattering on $ \mathbb{R}^d$ in even dimensions $ d \geq 2$ has many characteristics distinct from the odd-dimensional situation. In this article, we study the scattering matrix in even dimensions and prove several identities which hold for its meromorphic continuation onto $ \Lambda $, the Riemann surface of the logarithm function. We prove a theorem relating the multiplicities of the poles of the continued scattering matrix to the multiplicities of the poles of the continued resolvent. Moreover, we show that the poles of the scattering matrix on the $ m$th sheet of $ \Lambda $ are determined by the zeros of a scalar function defined on the physical sheet. Although analogs of these results are well known in odd dimension $ d$, we are unaware of a reference for all of $ \Lambda $ for the even-dimensional case. Our analysis also yields some surprising results about ``pure imaginary'' resonances. As an example, in contrast with the odd-dimensional case, we show that in even dimensions there are no ``pure imaginary'' resonances on any sheet of $ \Lambda $ for Schrödinger operators with potentials $ 0 \leq V \in L_0^\infty (\mathbb{R}^d)$.


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Additional Information

T. J. Christiansen
Affiliation: Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211-0001
Email: christiansent@missouri.edu

P. D. Hislop
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027
Email: peter.hislop@uky.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06458-1
Received by editor(s): December 6, 2013
Received by editor(s) in revised form: April 17, 2014
Published electronically: December 19, 2014
Additional Notes: The first author was partially supported by NSF grant DMS 1001156. The second author was partially supported by NSF grant 1103104.
Article copyright: © Copyright 2014 American Mathematical Society