An algebraic approach to a harmonic oscillator plus an inverse square potential in two dimensions

The exact solutions of the Schrödinger equation with a harmonic oscillator plus an inverse square potential are obtained in two dimensions. We construct the ladder operators directly from the radial wave functions and find that these operators satisfy the commutation relations of an SU(1, 1) group. We obtain the explicit expressions of the matrix elements for some related functions ρ and ρd/dρ with ρ = r². We also explore another symmetry between the eigenvalues E(r) and E(ir) by substituting r → ir.