Two families of superintegrable and isospectral potentials in two dimensions

Abstract

As an extension of the intertwining operator idea, an algebraic method which provides a link between supersymmetric quantum mechanics and quantum (super)integrability is introduced. By realization of the method in two dimensions, two infinite families of superintegrable and isospectral stationary potentials are generated. The method makes it possible to perform Darboux transformations in such a way that, in addition to the isospectral property, they acquire the superintegrability preserving property. Symmetry generators are second and fourth order in derivatives and all potentials are isospectral with one of the Smorodinsky-Winternitz potentials. Explicit expressions of the potentials, their dynamical symmetry generators and the algebra they obey as well as their degenerate spectra and corresponding normalizable states are presented.

Superintegrable quantum u(3) systems and higher rank factorizations

“…The figure on the right correspond to the the points of a q = 3 IUR of so (6). The three sections describe three IUR's of su (3).…”

“…Let H ℓ and H ℓ ′ be two Hamiltonians related by means of a differential operator X in the following form (6) where the triangular oposite faces correspond to two IUR's of su (3). The figure on the right correspond to the the points of a q = 3 IUR of so (6).…”

“…We also hope that this work will be useful when dealing with other integrable systems, but not necessarily maximally integrable, for instance not enjoing for such a wealth of factorizations, or even not having a system of separable variables, but still allowing for algebraic methods [2,3,4,5,6,7].…”

Intertwined Hamiltonians in two-dimensional curved spaces

“…It is shown that the solutions are related to Killing vector fields and isometry group of corresponding space. Our solutions are also closely related to the problem of superintegrability in 2D spaces [1,2,15,16].…”

“…The Intertwining relationship between differential operators is widely studied in recent years [1,2,3]. This is simply the following relationship between three differential operators H 1 , H 2 and L …”