Infinite-dimensional complex dynamics: A quantum random walk
Brendan Weickert
Discrete & Continuous Dynamical Systems - A 2001, 7(3): 517-524 doi: 10.3934/dcds.2001.7.517
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
A quantized henon map
John Erik Fornæss Brendan Weickert
Discrete & Continuous Dynamical Systems - A 2000, 6(3): 723-740 doi: 10.3934/dcds.2000.6.723
We quantize the classical Henon map on $\mathbb R^2$, obtaining a unitary map on $L^2 (\mathbb R)$ whose dynamics we study, developing analogies to the classical dynamics.
keywords: Hamiltonian. pseudodifferential operator Henon map

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