First eigenvalue characterization of Clifford hypersurfaces and Veronese surface

We give an estimate for the first eigenvalue of the Schrödinger operator $L:=-\mathrm{\Delta }-\sigma$ which is defined on the closed minimal submanifold $M^n$ in the unit sphere ${\mathbb{S}}^{n+m}$, where $\sigma$ is the square norm of the second fundamental form.