Energy eigenvalues of a quantum anharmonic oscillator from supersymmetry: the concept of conditional shape-invariance symmetry

Abstract

The eigenvalues for oscillators with quartic and sextic anharmonicities have been calculated for a restricted set of parameters by supersymmetric quantum mechanics. We show that energies of at most a few levels can be obtained algebraically for a severely restricted set of parameters resulting from the satisfaction of a few constraint conditions. We present how conditional shape-invariance symmetry of supersymmetric partner potentials leads to the conditional exact solution. We also indicate its generalization in N dimensions.