# American Institute of Mathematical Sciences

• Previous Article
Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential
• DCDS Home
• This Issue
• Next Article
Moment attractivity, stability and contractivity exponents of stochastic dynamical systems
July  2001, 7(3): 517-524. doi: 10.3934/dcds.2001.7.517

## Infinite-dimensional complex dynamics: A quantum random walk

 1 Dept of Mathematics, University of Chicago, Chicago, IL 60637, United States

Received  June 2000 Revised  January 2001 Published  April 2001

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.
Citation: Brendan Weickert. Infinite-dimensional complex dynamics: A quantum random walk. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 517-524. doi: 10.3934/dcds.2001.7.517
 [1] Håkon Hoel, Anders Szepessy. Classical Langevin dynamics derived from quantum mechanics. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 4001-4038. doi: 10.3934/dcdsb.2020135 [2] Xuwen Chen, Yan Guo. On the weak coupling limit of quantum many-body dynamics and the quantum Boltzmann equation. Kinetic and Related Models, 2015, 8 (3) : 443-465. doi: 10.3934/krm.2015.8.443 [3] John Erik Fornæss. Infinite dimensional complex dynamics: Quasiconjugacies, localization and quantum chaos. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 51-60. doi: 10.3934/dcds.2000.6.51 [4] Zhongyi Huang, Peter A. Markowich, Christof Sparber. Numerical simulation of trapped dipolar quantum gases: Collapse studies and vortex dynamics. Kinetic and Related Models, 2010, 3 (1) : 181-194. doi: 10.3934/krm.2010.3.181 [5] Paolo Antonelli, Seung-Yeal Ha, Dohyun Kim, Pierangelo Marcati. The Wigner-Lohe model for quantum synchronization and its emergent dynamics. Networks and Heterogeneous Media, 2017, 12 (3) : 403-416. doi: 10.3934/nhm.2017018 [6] François Gay-Balmaz, Cesare Tronci. Koopman wavefunctions and classical states in hybrid quantum–classical dynamics. Journal of Geometric Mechanics, 2022, 14 (4) : 559-596. doi: 10.3934/jgm.2022019 [7] José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2601-2617. doi: 10.3934/dcds.2020376 [8] Jianjun Yuan. Derivation of the Quintic NLS from many-body quantum dynamics in $T^2$. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1941-1960. doi: 10.3934/cpaa.2015.14.1941 [9] Helmut Kröger. From quantum action to quantum chaos. Conference Publications, 2003, 2003 (Special) : 492-500. doi: 10.3934/proc.2003.2003.492 [10] Alberto Ibort, Alberto López-Yela. Quantum tomography and the quantum Radon transform. Inverse Problems and Imaging, 2021, 15 (5) : 893-928. doi: 10.3934/ipi.2021021 [11] Paolo Antonelli, Pierangelo Marcati. Quantum hydrodynamics with nonlinear interactions. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 1-13. doi: 10.3934/dcdss.2016.9.1 [12] Luiza H. F. Andrade, Rui F. Vigelis, Charles C. Cavalcante. A generalized quantum relative entropy. Advances in Mathematics of Communications, 2020, 14 (3) : 413-422. doi: 10.3934/amc.2020063 [13] Gabriel Rivière. Remarks on quantum ergodicity. Journal of Modern Dynamics, 2013, 7 (1) : 119-133. doi: 10.3934/jmd.2013.7.119 [14] Sergei Avdonin, Pavel Kurasov. Inverse problems for quantum trees. Inverse Problems and Imaging, 2008, 2 (1) : 1-21. doi: 10.3934/ipi.2008.2.1 [15] Dmitry Jakobson. On quantum limits on flat tori. Electronic Research Announcements, 1995, 1: 80-86. [16] James B. Kennedy, Jonathan Rohleder. On the hot spots of quantum graphs. Communications on Pure and Applied Analysis, 2021, 20 (9) : 3029-3063. doi: 10.3934/cpaa.2021095 [17] Jin-Cheng Jiang, Chi-Kun Lin, Shuanglin Shao. On one dimensional quantum Zakharov system. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5445-5475. doi: 10.3934/dcds.2016040 [18] Dubi Kelmer. Quantum ergodicity for products of hyperbolic planes. Journal of Modern Dynamics, 2008, 2 (2) : 287-313. doi: 10.3934/jmd.2008.2.287 [19] Ruikuan Liu, Tian Ma, Shouhong Wang, Jiayan Yang. Thermodynamical potentials of classical and quantum systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1411-1448. doi: 10.3934/dcdsb.2018214 [20] Mason A. Porter, Richard L. Liboff. The radially vibrating spherical quantum billiard. Conference Publications, 2001, 2001 (Special) : 310-318. doi: 10.3934/proc.2001.2001.310

2021 Impact Factor: 1.588