(twenty decimal places) [@rr8; @rr9]. There is a factor of $(-1)^n$ in the sum. This unusual factor does not appear in conventional quantum mechanics. The presence of this factor is explained in the following discussion of orthonormality \[see (\[e8\])\] in which we encounter the problem associated with non-Hermitian $\mathcal{PT}$-symmetric Hamiltonians.

There seems to be a natural choice for the inner product of two functions $f(x)$ and $g(x)$: $$(f,g)\equiv\int dx\,[\mathcal{PT}f(x)]g(x),
\label{e7}$$ where $\mathcal{PT}f(x)=[f(-x)]^*$ and the integration path is the appropriate contour in the complex-$x$ plane. \[We will see that (\[e7\]) is [*not*]{} the correct choice for an inner product because it gives an indefinite metric. The correct inner product will be defined shortly, but studying this inner product is useful because it reveals the underlying mathematical structure of the theory.\] The apparent advantage of this inner product is that the associated norm $(f,f)$ is independent of the overall phase of $f(x)$ and is conserved in time because $H$ commutes with $\mathcal{PT}$ and the time-evolution operator is $e^{-iHt}$. Phase independence is desired because in quantum mechanics the objective is to construct a space of rays to represent quantum mechanical states. With respect to this inner product, the eigenfunctions $\phi_m(x)$ and $\phi_n(x)$ of 