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.
Categories: hep-th math-ph math.MP quant-ph