SUSY QM, symmetries and spectrum generating algebras for two-dimensional systems

Abstract

We show in a systematic and clear way how factorization methods can be used to construct the generators for hidden and dynamical symmetries. This is shown by studying the 2D problems of hydrogen atom, the isotropic harmonic oscillator and the radial potential Aρ 2ζ−2 − Bρ ζ−2 . We show that in these cases the non-compact (compact) algebra corresponds to so(2, 1) (su (2)).

su(1, 1) algebraic approach of the Dirac equation with Coulomb-type scalar and vector potentials in D+1 dimensions

“…In order to construct the su(1, 1) algebra generators, we apply the Schrödinger factorization [2,18] to the left-hand side of equation (12). Thus, we propose…”

Exact solutions of the Schrödinger equation for the pseudoharmonic potential: an application to some diatomic molecules

“…It should be noted that, Schrödinger factorization method has been less frequently applied to physical systems than Infeld-Hull factorization method, as it has been analyzed in detail by Martínez et al [33], and exemplified later with a typical system [32]. Martínez and Mota presented a systematic procedure of using the factorization method to construct the generators for hidden and dynamical symmetries, and applied this study to 2D problems of hydrogen atom, the isotropic harmonic and other radial potential of interest.…”

“…In a similar fashion, Salazar-Ramírez et al [34,35] have applied the factorization method to construct the generators of the dynamical algebra SU (1, 1) for the radial equation of the non-relativistic and relativistic generalized MICZ-Kepler problem. It should be noted that the generators in the examples [32][33][34][35] above have been constructed without adding phase as an additional variable like in Martínez-y-Romero et al [15]. Gur and Mann [30] have used the SU (1, 1) SGA method to construct the associated radial Barut-Girardello coherent states for the isotropic harmonic oscillator in arbitrary dimension and these states have been mapped into the Sturm-Coulomb radial coherent states.…”

Thesu(1, 1) dynamical algebra from the Schrödinger ladder operators forN-dimensional systems: hydrogen atom, Mie-type potential, harmonic oscillator and pseudo-harmonic oscillator

“…Since equation ( 15) is essentially the Schrödinger equation, we can immediately prove that D 3 is hermitian. Moreover, using equation ( 16) and (18) we can show…”