  For a given minimal Legendrian submanifold $L$ of a Sasaki-Einstein manifold
we construct two families of eigenfunctions of the Laplacian of $L$ and we give
a lower bound for the dimension of the corresponding eigenspace. Moreover, in
the case the lower bound is attained, we prove that $L$ is totally geodesic and
a rigidity result about the ambient manifold. This is a generalization of a
result for the standard Sasakian sphere done by L\^e and Wang.
