Level shift operators for open quantum systems

TL;DR

This paper investigates the intrinsic properties of level shift operators in open quantum systems at positive temperatures, revealing their role in resonance geometry, state relations, and ergodic behavior.

Contribution

It uncovers fundamental properties of level shift operators linked to system structure, resonance bifurcations, and ergodic properties, advancing understanding of open quantum systems at positive temperatures.

Findings

Level shift operators determine resonance geometry.

Degeneracy causes violation of Fermi Golden Rule.

Ergodic properties are affected by energy level degeneracy.

Abstract

Level shift operators describe the second order displacement of eigenvalues under perturbation. They play a central role in resonance theory and ergodic theory of open quantum systems at positive temperatures. We exhibit intrinsic properties of level shift operators, properties which stem from the structure of open quantum systems at positive temperatures and which are common to all such systems. They determine the geometry of resonances bifurcating from eigenvalues of positive temperature Hamiltonians and they relate the Gibbs state, the kernel of level shift operators, and zero energy resonances. We show that degeneracy of energy levels of the small part of the open quantum system causes the Fermi Golden Rule Condition to be violated and we analyze ergodic properties of such systems.