Milne quantization for non-Hermitian systems

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.

11 pages, 3 figures

Cited by in corpus (9)

• An introduction to PT-symmetric quantum mechanics -- time-dependent systems
• Controlling decoherence via $\mathcal{P}\mathcal{T}$-symmetric non-Hermitian open quantum systems
• Metric versus observable operator representation, higher spin models
• An introductory review on resource theories of generalized nonclassical light
• Relativistic Ermakov-Milne-Pinney Systems and First Integrals
• Exact solution for the time dependent non-Hermitian generalized Swanson oscillator
• Improved coherence time of a non-Hermitian qubit in a $\mathcal{P}\mathcal{T}$-symmetric Environment
• Effective size of a parity-time symmetric dimer
• Swanson Hamiltonian: non-PT-symmetry phase