$H$ in (\[e1\]) are orthogonal for $n\neq m$ because $H$ is symmetric. However, when we set $m=n$ we see by direct numerical calculation that the norm is evidently [*not positive*]{}: $$(\phi_m,\phi_n)=(-1)^n\delta_{mn}.
\label{e8}$$ This result is apparently true for all values of $\epsilon$ in (\[e3\]), and it has been verified numerically to extremely high precision. Because the norms of the eigenfunctions alternate in sign, the metric associated with the $\mathcal{PT}$ inner product $(\cdot,\cdot)$ is indefinite. This sign alternation appears to be a [*generic*]{} feature of this $\mathcal{PT}$ inner product. \[Extensive numerical calculations verify that the formula in (\[e8\]) holds for all $\epsilon\geq0$.\] We emphasize that while the sign of the norm of $\phi_n$ is hard to verify analytically, the orthogonality of $\phi_m$ and $\phi_n$ is a trivial consequence of the symmetry of $H$. It is necessary to assume that $H$ be symmetric in order to have this orthogonality.

In spite of the nonpositivity of the inner product, it is instructive to proceed with the usual analysis that one would perform for any Sturm-Liouville problem of the form $H\phi_n=E_n\phi_n$. First, we use the inner product formula (\[e8\]) to verify that (\[e6\]) is the representation of the unity operator. That is, we verify 