determined by the operator $e^{-iHt}$ whether the theory is expressed in terms of a $\mathcal{PT}$-symmetric Hamiltonian or just an ordinary Hermitian Hamiltonian. To establish the global unitarity of a theory we must show that as a state vector evolves, its norm does not change in time. If $\psi_0(x)$ is a prescribed initial wave function belonging to the Hilbert space spanned by the energy eigenstates, then it evolves into the state $\psi_t(x)$ at time $t$ according to $$\psi_t(x)=e^{-iHt}\psi_0(x).
\label{e19a}$$ With respect to the $\mathcal{CPT}$ inner product defined in (\[e18\]), the norm of the vector $\psi_t(x)$ does not change in time, $\langle\psi_t|\psi_t
\rangle=\langle\psi_0|\psi_0\rangle$, because the Hamiltonian $H$ commutes with the $\mathcal{CPT}$ operator.

Establishing unitarity at a local level is more subtle. Here, we must show that in coordinate space, there exists a local probability density that satisfies a continuity equation so that the probability does not leak away. This is a nontrivial consideration because the probability current flows about in the complex plane rather than along the real axis as in conventional Hermitian quantum mechanics. Preliminary numerical studies indeed indicate that the continuity equation is fulfilled [@rr12].

Just as states in the Schrödinger picture evolve in time according to the usual equation 