a metric of signature $(+,-)$. The condition $s^2>r^2\sin^2\theta$ ensures that $\mathcal{PT}$ symmetry is not broken. If this condition is violated, the states (\[e22\]) are no longer eigenstates of $\mathcal{PT}$ because $\alpha$ becomes imaginary. (When $\mathcal{PT}$ symmetry is broken, we find that the $\mathcal{PT}$ norm of the energy eigenstate vanishes.)

Next, we construct the operator $\mathcal{C}$: $$\mathcal{C}=\frac{1}{\cos\alpha}
\left(\begin{array}{cc} i\sin\alpha & 1 \cr 1 & -i\sin\alpha \end{array}\right).
\label{e23}$$ Note that $\mathcal{C}$ is distinct from $H$ and $\mathcal{P}$ and has the key property that $\mathcal{C}|\varepsilon_{\pm}\rangle=\pm|\varepsilon_{\pm}
\rangle$. The operator $\mathcal{C}$ commutes with $H$ and satisfies $\mathcal{C
}^2=1$. The eigenvalues of $\mathcal{C}$ are precisely the signs of the $\mathcal{PT}$ norms of the corresponding eigenstates. Using $\mathcal{C}$ we construct the new inner product structure $\langle u|v\rangle=(\mathcal{CPT}u)
\cdot v$. This inner product is positive definite because $\langle\varepsilon_{
\pm}|\varepsilon_{\pm}\rangle=1$. Thus, the two-dimensional Hilbert space spanned by $|\varepsilon_\pm\rangle$, with inner product $\langle\cdot|\cdot
\rangle$, has a Hermitian structure with signature $(+,+)$.

Let us demonstrate explicitly that the $\mathcal{CPT}$ norm of any vector is positive. For the arbitrary vector $\psi=\left({a\atop b}\right)$, where $a$ and $b$ are any complex numbers, we see that $\mathcal{T}\psi=\left(a^*\atop b^*
\right)$, that $\mathcal{PT}\psi=\left(b^*\atop a^*\right)$, and that $\mathcal{
CPT}\psi={1\over\cos\alpha}\,\left(a^*+ib^*\sin\alpha\atop b^*-ia^*\sin\alpha
\right)$. Thus, $\langle\psi|\psi\rangle=(\mathcal{CPT}\psi)\cdot\psi={1\over
\cos\alpha}[a^*a+b^*b+i(b^*b-a^*a)\sin\alpha]$. Now let $a=x+iy$ and 