New results for Heisenberg dynamics for non self-adjoint Hamiltonians

TL;DR

This paper investigates Heisenberg dynamics in quantum systems with non self-adjoint Hamiltonians, emphasizing the importance of normalized vectors and conditions for conserved quantities, extending previous analyses in this emerging research area.

Contribution

It explores the role of normalized vectors in Heisenberg dynamics and identifies conditions for conserved observables in systems with non self-adjoint Hamiltonians.

Findings

Normalized vectors are crucial for analyzing Heisenberg dynamics.

Conditions for observables to remain constant over time are identified.

The study extends understanding of non self-adjoint Hamiltonian systems.

Abstract

In a previous paper we began our analysis on the role of non self-adjoint Hamiltonians in connection with the Heisenberg dynamics for quantum systems. Here, motivated by the growing interest on this topic and on some recent results on dynamical systems, we continue this analysis focusing on what we believe is an unexplored (or, at least, not so explored! aspect of Heisenberg dynamics, related to the need for using vectors which are {\em brute-force normalized}. Our main interest is on conserved quantities, and on conditions which guarantee that some observables of the system, or their mean values, do not evolve in time.