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We consider Toeplitz operators associated with the renormalizedBochner-Laplacian on high tensor powers of a positive line bundle on a compactsymplectic manifold. We study the asymptotic behavior, in the semiclassicallimit, of low-lying eigenvalues and the corresponding eigenfunctions of aself-adjoint Toeplitz operator under assumption that its principal symbol has anon-degenerate minimum with discrete wells. As an application, we prove upperbounds for low-lying eigenvalues of the Bochner-Laplacian in the semiclassicallimit.

Semiclassical spectral analysis of Toeplitz operators on symplectic manifolds: the case of discrete wells