  We consider Toeplitz operators associated with the renormalized
Bochner-Laplacian on high tensor powers of a positive line bundle on a compact
symplectic manifold. We study the asymptotic behavior, in the semiclassical
limit, of low-lying eigenvalues and the corresponding eigenfunctions of a
self-adjoint Toeplitz operator under assumption that its principal symbol has a
non-degenerate minimum with discrete wells. As an application, we prove upper
bounds for low-lying eigenvalues of the Bochner-Laplacian in the semiclassical
limit.
