We formulate quantum mechanics for non-Hermitian Hamiltonians that are invariant under $\mathcal{P}\mathcal{T}$, where $\mathcal{P}$ is the parity and $\mathcal{T}$ denotes time reversal, for the case that time-reversal symmetry is odd (${\mathcal{T}}^2=-1$), generalizing prior work for the even case (${\mathcal{T}}^2=1$). We discover an analog of Kramer’s theorem for $\mathcal{P}\mathcal{T}$ quantum mechanics, present a prototypical example of a $\mathcal{P}\mathcal{T}$ quantum system with odd time reversal, and discuss potential applications of the formalism.

• Received 31 August 2009

DOI:https://doi.org/10.1103/PhysRevA.82.042101