A new characterization of submanifolds with parallel mean curvature vector in $S^{n+p}$

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In this work we will consider compact submanifold $M^{n}$ immersed in the Euclidean sphere $S^{n+p}$ with parallel mean curvature vector and we introduce a Schr\{o}dinger operator $L=-\Delta+V$, where $\Delta$ 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 $\mu_{1}$ of $L$, by showing that either $\mu_{1}=0$ or $\mu_{1}\leq-n(1+H^{2})$. As a consequence we obtain new characterizations of spheres, Clifford tori and Veronese surfaces that extend a work due to Wu \cite{wu} for minimal submanifolds.