point $x=-1$ rather than $x=0$. See Ref. [@rr2] for the general construction of the relevant parity operator.

With properly defined boundary conditions the spectrum of the Hamiltonian $H$ in (\[e1\]) is [*real and positive*]{} when $\epsilon\geq0$ [@rr3] and the spectrum is partly real and partly complex when $\epsilon<0$. The eigenvalues have been computed numerically to very high precision, and the real eigenvalues are plotted as functions of $\epsilon$ in Fig. \[f1\].

The $\mathcal{PT}$ symmetry of a Hamiltonian $H$ is said to be [*unbroken*]{} if all of the eigenfunctions of $H$ are simultaneously eigenfunctions of $\mathcal{
PT}$. Note that even if a system is defined by an equation that possesses a discrete symmetry, the solution to this equation need not exhibit that symmetry. For example, although the differential equation ${\ddot y}(t)=y(t)$ is symmetric under time reversal $t\to-t$, the solutions $y(t)=e^t$ and $y(t)=e^{-t}$ do not exhibit time-reversal symmetry; other solutions, such as $y(t)=\cosh(t)$, are time-reversal symmetric. The same is true of a system whose Hamiltonian is $\mathcal{PT}$ symmetric. Even if the Schrödinger equation and corresponding boundary conditions are $\mathcal{PT}$ symmetric, the wave function that solves the Schrödinger equation boundary value problem need not be symmetric under space-time reflection. When the solution exhibits 