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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Spectral properties and dynamics of quantized Henon maps


Author: Brendan Weickert
Journal: Trans. Amer. Math. Soc. 356 (2004), 4951-4968
MSC (2000): Primary 32H50; Secondary 37N20
DOI: https://doi.org/10.1090/S0002-9947-04-03475-0
Published electronically: April 16, 2004
MathSciNet review: 2084407
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Abstract: We study a generalization of the Airy function, and use its properties to investigate the dynamics and spectral properties of the unitary operators on $L^2(\mathbf{R})$ of the form $U_c:=Fe^{i(q(x)+cx)}$, where $q$ is a real polynomial of odd degree, $c$ is a real number, and $F$ is the Fourier transform. We show that $U_c$ is a quantization of the classical Henon map
\begin{align*}f_\lambda:\mathbf{R}^2 &\to \mathbf{R}^2 ,\\ (x,y) &\mapsto (y+q'(x)+c,-x), \end{align*}
and show that for $c>0$ sufficiently large, $U_c$ has purely continuous spectrum. This fact has implications for the dynamics of $U_c$, which are shown to correspond when the condition is satisfied to the dynamics of its classical counterpart on $\mathbf{R}^2$.


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Additional Information

Brendan Weickert
Affiliation: Department of Mathematics, Washington and Lee University, Lexington, Virginia 24450
Email: weickertb@wlu.edu

DOI: https://doi.org/10.1090/S0002-9947-04-03475-0
Received by editor(s): January 15, 2003
Received by editor(s) in revised form: July 3, 2003
Published electronically: April 16, 2004
Article copyright: © Copyright 2004 American Mathematical Society