Exact Isospectral Pairs of PT-Symmetric Hamiltonians

Abstract

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 $\hslash$). 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.

arXiv comment18 pages, 12 figures

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