Heisenberg dynamics for non self-adjoint Hamiltonians: symmetries and derivations

TL;DR

This paper explores the dynamics of non self-adjoint Hamiltonians in quantum mechanics using a Heisenberg-like framework, highlighting differences from standard cases and analyzing symmetries and conserved quantities.

Contribution

It introduces a Heisenberg-like approach for non self-adjoint Hamiltonians, emphasizing the role of symmetries, *-derivations, and integrals of motion, which is a novel perspective.

Findings

Differences between Heisenberg dynamics for self-adjoint and non self-adjoint Hamiltonians.

Identification of symmetries and *-derivations in non self-adjoint systems.

Discussion on conserved quantities in the context of non self-adjoint Hamiltonians.

Abstract

In some recent literature the role of non self-adjoint Hamiltonians, $H\neq H^\dagger$, is often considered in connection with gain-loss systems. The dynamics for these systems is, most of the times, given in terms of a Schr\"odinger equation. In this paper we rather focus on the Heisenberg-like picture of quantum mechanics, stressing the (few) similarities and the (many) differences with respected to the standard Heisenberg picture for systems driven by self-adjoint Hamiltonians. In particular, the role of the symmetries, *-derivations and integrals of motion is discussed.