Comment on “Reply to Comment on Time-dependent Quasi-Hermitian Hamiltonians and the Unitary Quantum Evolution”

I point out that if one defines the operator UR(t) as done by M. Znojil in his reply [arXiv:0711.0514v1] to my comment [arXiv:0711.0137v1] and also accepts the validity of the defining relation of UR(t) as given in his paper [arXiv:0710.5653v1], one finds that the time-evolution of the associated quantum system is not governed by the Schrödinger equation for the Hamiltonian operator H but an operator H ′ which differs from H if the metric operator is time-dependent. In the latter case this effective Hamiltonian H ′ is not observable. This is consistent with the conclusions of my paper [Phys. Lett. B 650, 208 (2007), arXiv:0706.1872v2] which allow for unitary time-evolution generated by unobservable Hamiltonians. In [1] M. Znojil explains that my comment [2] on his paper [3] is not relevant, because it relies on the relations i~∂tUR(t) = H(t)UR(t), UR(0) = I, (1) which he identifies as an “incorrect” assumption. To clarify the matter, recall that UR(t) was initially introduced in Eq. (14) of [3] which reads |Φ(t)〉 = UR(t)|Φ(0)〉, (2) where |Φ(t)〉 is an arbitrary evolving state vector. Clearly (1) follows from (2), if one postulates the standard Schrödinger time-evolution defined by the Hamiltonian operator H , namely i~∂t|Φ(t)〉 = H(t)|Φ(t)〉. (3)