Infinite-dimensional complex dynamics: A quantum random walk

Pages: 517 - 524, Volume 7, Issue 3, July 2001

doi:10.3934/dcds.2001.7.517       Abstract        Full Text (181.6K)       Related Articles

Brendan Weickert - Dept of Mathematics, University of Chicago, Chicago, IL 60637, United States (email)

Abstract: We describe a unitary operator $U(\alpha)$ on L²$(\mathbb T)$, depending on a real parameter $\alpha$, that is a quantization of a simple piecewise holomorphic dynamical system on the cylinder $\mathbf C^* \cong \mathbb T \times \mathbb R$. We give results describing the spectrum of $U(\alpha)$ in terms of the diophantine properties of $\alpha$, and use these results to compare the quantum to classical dynamics. In particular, we prove that for almost all $\alpha$, the quantum dynamics localizes, whereas the classical dynamics does not. We also give a condition implying that the quantum dynamics does not localize.

Keywords:  Quantum dynamics
Mathematics Subject Classification:  Primary: 37N20.

Received: June 2000;      Revised: January 2001;      Published: April 2001.