Two Dimensional Hamiltonian with Generalized Shape Invariance Symmetry

Abstract

The two dimensional Hamiltonian with generalized shape invariance symmetry over $S^2$, has been obtained via Fourier transformation over the three coordinates of the $SU(3)$ Casimir operator defined on $SU(3)/SU(2)$ symmetric space. It is shown that the generalized shape invariance is equivalent to $SU(3)$ symmetry and that there is one to one correspondence between the representations of the generalized shape invariance and $SU(3)$ Verma modules. Also the two dimensional Hamiltonian in $\mathbb{R}^2$ space which posseses ordinary shape invariance symmetry with respect to two parameters, has been obtained via Inönü–Wigner contraction over $SU(3)$ manifold.