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Milne quantization for non-Hermitian systems

Fring, A., Dey, S. & Gouba, L. (2015). Milne quantization for non-Hermitian systems. Journal of Physics A: Mathematical and Theoretical, 48, 40. doi: 10.1088/1751-8113/48/40/40FT01


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.

Publication Type: Article
Additional Information: This is an author-created, un-copyedited version of an article accepted for publication in Journal of Physics A: Mathematical and Theoretical. The publisher is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at
Subjects: Q Science > QA Mathematics
Departments: School of Science & Technology > Mathematics
Text - Accepted Version
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