Hamiltonian formulation of linear non-Hermitian systems

TL;DR

This paper develops a Hamiltonian framework for linear non-Hermitian systems, enabling the identification of conserved quantities and adiabatic invariants, and unifies the treatment with Hermitian quantum mechanics.

Contribution

It introduces a Hamiltonian formulation for non-Hermitian systems, extending classical mechanics concepts to these systems and linking them with known Hermitian results.

Findings

Hamiltonian can be constructed for non-Hermitian systems

Conserved charge identified via Noether's theorem

Adiabatic invariants recognized in the formulation

Abstract

For a linear non-Hermitian system, I demonstrate that a Hamiltonian can be constructed such that the non-Hermitian equations can be expressed exactly in the form of Hamilton's canonical equations. This is first shown for discrete systems and then extended to continuous systems. With this Hamiltonian formulation, I am able to identify a conserved charge by applying Noether's theorem and recognize adiabatic invariants. When applied to Hermitian systems, all the results reduce to the familiar ones associated with the Schr\"odinger equation.