  We study the near diagonal asymptotic expansion of the generalized Bergman
kernel of the renormalized Bochner-Laplacian on high tensor powers of a
positive line bundle over a compact symplectic manifold. We show how to compute
the coefficients of the expansion by recurrence and give a closed formula for
the first two of them. As consequence, we calculate the density of states
function of the Bochner-Laplacian and establish a symplectic version of the
convergence of the induced Fubini-Study metric. We also discuss generalizations
of the asymptotic expansion for non-compact or singular manifolds as well as
their applications. Our approach is inspired by the analytic localization
techniques of Bismut-Lebeau.
