Milne quantization for non-Hermitian systems

Abstract

We generalize the Milne quantization condition to non-Hermitian systems. In\nthe general case the underlying nonlinear Ermakov-Milne-Pinney equation needs\nto be replaced by a nonlinear integral differential equation. However, when the\nsystem is PT-symmetric or/and quasi/pseudo-Hermitian the equations simplify and\none may employ the original energy integral to determine its quantization. We\nillustrate the working of the general framework with the Swanson model and two\nexplicit examples for pairs of supersymmetric Hamiltonians. In one case both\npartner Hamiltonians are Hermitian and in the other a Hermitian Hamiltonian is\npaired by a Darboux transformation to a non-Hermitian one.\n