González López, ArtemioTanaka, Toshiaki2023-06-202023-06-202006-04-070305-447010.1088/0305-4470/39/14/014https://hdl.handle.net/20.500.14352/51544©IOP Publishing. This work was partially supported by Spain’s DGI under the grant No. BFM2002-02646 (AG-L) as well as by the National Science Council of the Republic of China under the grant No. NSC-93-2112-M-032-009 (TT).We recall the importance of recognizing the different mathematical nature of various concepts relating to PT -symmetric quantum theories. After clarifying the relation between supersymmetry and pseudo-supersymmetry, we prove generically that nonlinear pseudo-supersymmetry, recently proposed by Sinha and Roy, is just a special case of N -fold supersymmetry. In particular, we show that all the models constructed by these authors have type A 2-fold supersymmetry. Furthermore, we prove that an arbitrary one-body quantum Hamiltonian which admits two (local) solutions in closed form belongs to type A 2-fold supersymmetry, irrespective of whether or not it is Hermitian, PT -symmetric, pseudo-Hermitian, and so on.engjournal articlehttp://dx.doi.org/10.1088/0305-4470/39/14/014http://iopscience.iop.orghttp://arxiv.org/abs/quant-ph/0602177open access51-73Non-hermitian hamiltoniansQuantum-mechanicsPt-symmetryPeriodic potentialsReal spectraSystemsTransformationsFísica-Modelos matemáticosFísica matemática