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

TL;DR

This paper develops a quantum version of the Spectral Decomposition Theorem using the Wigner transform, enabling analysis of quantum ergodic hierarchy levels, classical limits, and spectrum connections, with applications to microwave billiards and Gamow models.

Contribution

It introduces the Quantum Spectral Decomposition Theorem (QSDT), linking quantum ergodic hierarchy levels to spectral properties and classical limits.

Findings

QSDT successfully characterizes quantum ergodic hierarchy levels.

The connection between quantum spectrum and ergodic hierarchy is established.

Applications demonstrate the physical relevance of QSDT in real systems.

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].