of these Hamiltonians have also been shown to be real and positive [@rr5].

It is important to know whether a given non-Hermitian $\mathcal{PT}$-symmetric Hamiltonian has a positive real spectrum, but the most urgent question is whether such a Hamiltonian defines a physical theory of quantum mechanics. By a [*physical theory*]{} we mean that there is a Hilbert space of state vectors and that this Hilbert space has an inner product with a positive norm. In quantum mechanics we interpret the norm of a state as a probability and this probability must be positive. Furthermore, we must show that the time evolution of the theory is unitary. This means that as a state vector evolves in time the probability does not leak away. With these considerations in mind one would wonder whether a Hamiltonian such as $H$ in (\[e1\]) gives a consistent quantum theory. Indeed, early investigations of this Hamiltonian have shown that while the spectrum is entirely real and positive when $\epsilon\geq0$, one inevitably encountered the severe problem of a Hilbert space endowed with an indefinite metric [@rr6].

However, there is a new symmetry that all $\mathcal{PT}$-symmetric Hamiltonians having an unbroken $\mathcal{PT}$-symmetry possess [@rr7]. We denote the operator representing this 