Milne quantization for non-Hermitian systems

TL;DR

This paper extends the Milne quantization method to non-Hermitian quantum systems, simplifying the process for PT-symmetric and quasi-Hermitian cases, and demonstrates it with specific models and Hamiltonian pairs.

Contribution

It introduces a generalized Milne quantization framework for non-Hermitian systems, reducing to simpler forms under PT-symmetry or pseudo-Hermiticity, with practical examples.

Findings

The generalized framework applies to PT-symmetric and quasi-Hermitian systems.

Simplified equations enable easier quantization of non-Hermitian models.

Illustrative examples include the Swanson model and supersymmetric Hamiltonians.

Abstract

We generalize the Milne quantization condition to non-Hermitian systems. In the general case the underlying nonlinear Ermakov-Milne-Pinney equation needs to be replaced by a nonlinear integral differential equation. However, when the system is PT-symmetric or/and quasi/pseudo-Hermitian the equations simplify and one may employ the original energy integral to determine its quantization. We illustrate the working of the general framework with the Swanson model and two explicit examples for pairs of supersymmetric Hamiltonians. In one case both partner Hamiltonians are Hermitian and in the other a Hermitian Hamiltonian is paired by a Darboux transformation to a non-Hermitian one.