Application of conditional shape invariance symmetry to obtain the eigen-spectrum of the mixed potential V(r)=ar+br2+cr+l(l+1)r2

- Conditional shape invariance symmetry of mixed potential is proved.
- Exploiting CES symmetry, excited energy states are calculated.
- Analytical and numerical results are compared.
- To obtain eigen spectrum of CES potential a new methodology is presented.

We show that the conditional shape invariance symmetry can be used as a very powerful tool to calculate the eigenvalues of the mixed potential V(r)=ar+br2+cr+l(l+1)r2 for a restricted set of potential parameters. The energy for any state can be obtained algebraically, albeit for a severely restricted set of potential parameters. We also indicate that each member of the hierarchy of Hamiltonians is basically conditionally translational shape invariant. Comparison of analytically obtained results with numerical results is also presented. Our present methodology can be taken as an alternative treatment for the calculation of any higher order excited states of conditionally exactly solvable (CES) potentials.