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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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Spectral properties and dynamics of quantized Henon maps
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by Brendan Weickert PDF
Trans. Amer. Math. Soc. 356 (2004), 4951-4968 Request permission

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
  • Received by editor(s): January 15, 2003
  • Received by editor(s) in revised form: July 3, 2003
  • Published electronically: April 16, 2004
  • © Copyright 2004 American Mathematical Society
  • 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
  • MathSciNet review: 2084407