Abstract: We estimated the upper bound of the first eigenvalue of the Schrdinger operator L=-Δ-k(2-1/p)(S-nH²) on the ndimensional closed kextremal (k≥1) submanifold Mⁿ in a unit sphere  S^n+p(n≥3) by choosing suitable text function. Then, we gave some characteristics of submanifolds M^n based on the eigenvalue, where H and S were the mean curvature and the squared length of the second fundamental form, of Mⁿ respectively, Δ was the Laplace operator on Mⁿ.

Key words: first eigenvalue, Schrodinger operator, k-extremal submanifold