Shape invariant potentials in higher dimensions

Cyclic Shape Invariant Potentials

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 osc...

Shape matching in higher dimensions

Shape Invariant Potentials in “Discrete Quantum Mechanics”

Shape invariance is an important ingredient of many exactly solvable quantum mechanics. Several examples of shape invariant “discrete quantum mechanical systems” are introduced and discussed in some detail. They arise in the problem of describing the equilibrium positions of Ruijsenaars-Schneider type systems, which are “discrete” counterparts of Calogero and Sutherland systems, the celebrated ...

Vanishing Scalar Invariant Spacetimes in Higher Dimensions

We study manifolds with Lorentzian signature and prove that all scalar curvature invariants of all orders vanish in a higher-dimensional Lorentzian spacetime if and only if there exists an aligned non-expanding, non-twisting, geodesic null direction along which the Riemann tensor has negative boost order.

Barrier Penetration for Supersymmetric Shape-Invariant Potentials

Exact reflection and transmission coefficients for supersymmetric shapeinvariant potentials barriers are calculated by an analytical continuation of the asymptotic wave functions obtained via the introduction of new generalized ladder operators. The general form of the wave function is obtained by the use of the F(−∞,+∞)-matrix formalism of Fröman and Fröman which is related to the evolution of...