closed-form expression for the [*spectral zeta function*]{} (sum of the inverses of the energy eigenvalues). To do so we set $z=x$ in $G(x,z)$ and use (\[e8\]) to integrate over $x$. For all $\epsilon>0$ we obtain [@rr9] $$\sum_n {1\over E_n}=\left[1+{\cos\left({3\epsilon\pi\over2\epsilon+8}\right)\sin
\left({\pi\over4+\epsilon}\right)\over\cos\left({\epsilon\pi\over4+2\epsilon}
\right)\sin\left({3\pi\over4+\epsilon}\right)}\right]{\Gamma\left({1\over4+
\epsilon}\right)\Gamma\left({2\over4+\epsilon}\right)\Gamma\left({\epsilon\over
4+\epsilon}\right)\over(4+\epsilon)^{4+2\epsilon\over4+\epsilon}\Gamma\left({1+
\epsilon\over4+\epsilon}\right)\Gamma\left({2+\epsilon\over4+\epsilon}\right)}.
\label{e13}$$ This result has been verified to extremely high numerical accuracy [@rr9].

All of these general Sturm-Liouville constructions are completely standard. But now we must address the crucial question of whether a $\mathcal{PT}$-symmetric Hamiltonian defines a physically viable quantum mechanics or whether it merely provides an amusing Sturm-Liouville eigenvalue problem. The apparent difficulty with formulating a quantum theory is that the vector space of quantum states is spanned by energy eigenstates, of which half have $\mathcal{PT}$ norm $+1$ and half have $\mathcal{PT}$ norm $-1$. Because the norm of the states carries a probabilistic interpretation in standard quantum theory, the existence of an indefinite metric in (\[e8\]) seems to be a serious obstacle. The situation here in which half of the energy eigenstates have positive norm and half have negative norm is analogous to the problem that Dirac encountered in formulating the spinor wave equation in relativistic quantum theory [@rr10]. Following Dirac’s approach, we attack the problem 