Cyclic Shape Invariant Potentials

Abstract

We formulate and study the set of coupled nonlinear differential equations which define a series of shape invariant potentials which repeats after a cycle of $p$ iterations. These cyclic shape invariant potentials enlarge the limited reservoir of known analytically solvable quantum mechanical eigenvalue problems. At large values of $x$, cyclic superpotentials are found to have a linear harmonic oscillator behavior with superposed oscillations consisting of several systematically varying frequencies. At the origin, cyclic superpotentials vanish when the period $p$ is odd, but diverge for $p$ even. The eigenvalue spectrum consists of $p$ infinite sets of equally spaced energy levels, shifted with respect to each other by arbitrary energies $\omega_0,\omega_1,\...,\omega_{p-1}$. As a special application, the energy spacings $\omega_k$ can be identified with the periodic points generatedby the logistic map $z_{k+1}=r z_k (1 - z_k)$. Increasing the value of $r$ and following the bifurcation route to chaos corresponds to studying cyclic shape invariant potentials as the period $p$ takes values 1,2,4,8,...