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.