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 ${\omega }_0,{\omega }_1,\dot{.}.,{\omega }_{p-1}$. 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,...

13 pages, 4 figures

References in corpus (5)

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• Exactly Solvable Potentials and Quantum Algebras
• New Shape Invariant Potentials in Supersymmetric Quantum Mechanics
• Potentials with Two Shifted Sets of Equally Spaced Eigenvalues and Their Calogero Spectrum

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