A Quantum Version of The Spectral Decomposition Theorem of Dynamical Systems, Quantum Chaos Hierarchy: Ergodic, Mixing and Exact

Abstract

In this paper we study Spectral Decomposition Theorem [1] and translate it to quantum language by means of the Wigner transform. We obtain a quantum version of Spectral Decomposition Theorem (QSDT) which enables us to achieve three distinct goals: First, to rank Quantum Ergodic Hierarchy levels [2,3]. Second, to analyze the classical limit in quantum ergodic systems and quantum mixing systems. And third, and maybe most important feature, to find a relevant and simple connection between the first three levels of quantum ergodic hierarchy (ergodic, exact and mixing) and quantum spectrum. Finally, we illustrate the physical relevance of QSDT applying it to two examples: Microwave billiards [4,5] and a phenomenological Gamow model type [6,7].

Chaos, Solitons & Fractals (2014)

References in corpus (2)

• Towards a definition of the Quantum Ergodic Hierarchy: Kolmogorov and Bernoulli systems
• Non-Hermitian Hamilton operator in open quantum systems

Cited by in corpus (6)

• Notions of the ergodic hierarchy for curved statistical manifolds
• Lyapunov exponents and Hamiltonian poles in a non Hermitian dynamics
• Majorization and dynamics of continuous distributions
• Distinguishability notion based on Wootters statistical distance: application to discrete maps
• An upper bound for the KS-entropy in quantum mixing systems
• An estimation of the logarithmic timescale in an ergodic dynamics