Exact isospectral pairs of symmetric Hamiltonians

Abstract

A technique for constructing an infinite tower of pairs of symmetric Hamiltonians, and (n = 2, 3, 4, …), that have exactly the same eigenvalues is described and illustrated by means of three examples (n = 2, 3, 4). The eigenvalue problem for the first Hamiltonian 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 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 is real, the Hamiltonian exhibits quantum anomalies (terms proportional to powers of ℏ). These anomalies are remnants of the complex nature of the equivalent Hamiltonian . For the cases n = 2, 3, 4 in the classical limit in which the anomaly terms in are discarded, the pair of Hamiltonians Hn,classical and Kn,classical have closed classical orbits whose periods are identical.