In this work we will consider compact submanifold M n immersed in the Euclidean sphere S nþp with parallel mean curvature vector and we introduce a Schrö dinger operator L ¼ ÀD þ V , where D stands for the Laplacian whereas V is some potential on M n which depends on n; p and h that are respectively, the dimension, codimension and mean curvature vector of M n . We will present a gap estimate for the first eigenvalue m 1 of L, by showing that eitherAs a consequence we obtain new characterizations of spheres, Cli¤ord tori and Veronese surfaces that extend a work due to Wu [W] for minimal submanifolds.

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