Singularity in Potential, Perturbation and Variational Methods

In this work, we carefully study the energy eigen-values and splitting of heavy quarkonia as there exist $1/r^3$ and $\delta^3(\vec r)$ singular terms in the potential which make a direct numerical solution of the Schr\"{o}dinger equation impossible. We compare the results obtained in terms of perturbation and variational methods with various treatments.