The Magnus expansion for non-Hermitian Hamiltonians

TL;DR

This paper develops a generalized Magnus expansion that preserves unitarity for all bounded finite-dimensional non-Hermitian Hamiltonians, extending the method's applicability beyond Hermitian cases.

Contribution

It introduces a new form of the Magnus expansion that guarantees unitarity for non-Hermitian Hamiltonians, a property not preserved in standard expansions.

Findings

The generalized expansion maintains unitarity for bounded finite-dimensional non-Hermitian Hamiltonians.

The method extends the theoretical framework of the Magnus expansion to non-Hermitian systems.

It provides a tool for analyzing time evolution in non-Hermitian quantum mechanics.

Abstract

The Magnus expansion offers a method to express a time-ordered exponential as an ordinary operatorial exponential. This representation has advantageous theoretical properties, while still solving the original differential equation. For any finite dimensional Hermitian Hamiltonians, the standard Magnus expansion guarantees a manifestly unitary representation. However, this property is no longer preserved if the Hamiltonian is infinite dimensional or non-Hermitian. In this work, we derive a generalized expansion that maintains the property of unitarity for all bounded finite dimensional Hamiltonians.