Comment on “Time-dependent quasi-Hermitian Hamiltonians and the unitary quantum evolution”

In arXiv:0710.5653v1 M. Znojil claims that he has found and corrected an error in my paper: [Phys. Lett. B 650, 208 (2007), arXiv:0706.1872v2] and that it is possible to escape its main conclusion, namely that the unitarity of the time-evolution and observability of the Hamiltonian imply time-independence of the metric operator. In this note I give a very short calculation showing that the analysis given by M. Znojil also leads to the same conclusion as of my above-mentioned paper. This is actually a reconfirmation of the validity of the results of the latter paper. PACS number: 03.65.Ca, 11.30.Er, 03.65.Pm, 11.80.Cr First I recall the notation used in [1]. • Θ is a possibly time-dependent (positive) metric operator. • H is a possibly time-dependent Θ-pseudo-Hermitian Hamiltonian operator acting on a reference Hilbert space H with the inner product 〈·|·〉, i.e., H(t) = Θ(t)H(t)Θ(t). (1) • ω := √ Θ and h := ωHω is the equivalent Hermitian Hamiltonian with evolution operator u, i.e., i~∂tu(t) = h(t)u(t) and u(0) = I, where I stands for the identity operator. • UR is the evolution operator for H , i.e., i~∂tUR(t) = H(t)UR(t), UR(0) = I. (2)