  A technique for constructing an infinite tower of pairs of PT-symmetric
Hamiltonians, $\hat{H}_n$ and $\hat{K}_n$ (n=2,3,4,...), that have exactly the
same eigenvalues is described. The eigenvalue problem for the first Hamiltonian
$\hat{H}_n$ of the pair must be posed in the complex domain, so its
eigenfunctions satisfy a complex differential equation and fulfill homogeneous
boundary conditions in Stokes' wedges in the complex plane. The eigenfunctions
of the second Hamiltonian $\hat{K}_n$ of the pair obey a real differential
equation and satisfy boundary conditions on the real axis. This equivalence
constitutes a proof that the eigenvalues of both Hamiltonians are real.
Although the eigenvalue differential equation associated with $\hat{K}_n$ is
real, the Hamiltonian $\hat{K}_n$ exhibits quantum anomalies (terms
proportional to powers of $\hbar$). These anomalies are remnants of the complex
nature of the equivalent Hamiltonian $\hat{H}_n$. In the classical limit in
which the anomaly terms in $\hat{K}_n$ are discarded, the pair of Hamiltonians
$H_{n,classical}$ and $K_{n,classical}$ have closed classical orbits whose
periods are identical.
