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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 ω 0 , ω 1 , . . . , ω p−1 . As a special application, the energy spacings ω k can be identified with the periodic points generated by the logistic map z k+1 = rz 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, . . .

Abstract:We review solvable models within the framework of supersymmetric quantum mechanics (SUSYQM). In SUSYQM, the shape invariance condition insures solvability of quantum mechanical problems. We review shape invariance and its connection to a consequent potential algebra. The additive shape invariance condition is specified by a difference-differential equation; we show that this equation is equivalent to an infinite set of partial differential equations. Solving these equations, we show that the known list of -independent superpotentials is complete. We then describe how these equations could be extended to include superpotentials that do depend on .

We give a systematic classification and a detailed discussion of the structure, motion and scattering of the recently discovered negaton and positon solutions of the Korteweg-de Vries equation. There are two distinct types of negaton solutions which we label [S n ] and [C n ], where (n + 1) is the order of the Wronskian used in the derivation. For negatons, the number of singularities and zeros is finite and they show very interesting time dependence. The general motion is in the positive x direction, except for certain negatons which exhibit one oscillation around the origin. In contrast, there is just one type of positon solution, which we label [C n ]. For positons, one gets a finite number of singularities for n odd, but an infinite number for even values of n. The general motion of positons is in the negative x direction with periodic oscillations. Negatons and positons retain their identities in a scattering process and their phase shifts are discussed. We obtain a simple explanation of all phase shifts by generalizing the notions of "mass" and "center of mass" to singular solutions. Finally, it is shown that negaton and positon solutions of the KdV equation can be used to obtain corresponding new solutions of the modified KdV equation.