Estimates on the Lower Bound of the First Gap

We give a new lower bound for the first gap ${\lambda }_2-{\lambda }_1$ of the Dirichlet eigenvalues of the Schr{"o}dinger operator on a bounded convex domain $\mathrm{\Omega }$ in R${}^n$ or S${}^n$ and greatly sharpens the previous estimates. The new bound is explicit and computable.