Our recent post "On non-Hermitian quantum mechanics" arXiv:0804.2051 we critique some recent work in the field by Carl Bender et al [see for example Phys. Rev. Lett 98, 040403 (2007) and Rep. Prog. Phys. 70, 947-1018 (2007)]. Their work is of interest but, in our opinion, there is some misuse of terminology with respect to the label "non-Hermitian". To us it is more like they are doing conventional quantum mechanics with novel inner products.
There already exists in the literature comments on this topic (e.g. Ali Mostafazadeh arXiv:quant-ph/0310164 and arXiv:quant-ph/0407070 as well as David B. Fairlie and Jean Nuyts arXiv:hep-th/0412148). While these papers present valid arguments they miss the central point of our comment. Namely, that if one chooses such an inner product then the Hamiltonian in question is actually Hermitian (and the whole exercise is somewhat redundant). Our argument is based on an observation that there exists in the literature some confusion over the definition of Hermiticity and the imposition of Schrödinger evolution on the evolution of state vectors. [This argument is not quite the same as the one presented in L.D. Landau and E.M. Lifshitz, “Quantum Mechanics (Non-relativistic Theory)” which, as we point out, also contains a flaw].