Partial Bergman kernels $\Pi_{ k, E}$ are kernels of orthogonal projections onto subspaces $\mathcal{ k} \subset H^0(M, L^k)$ of holomorphic sections of the $k$th power of an ample line bundle over a Kahler manifold $(M, \omega)$. The subspaces of this article are spectral subspaces $\{ \hat{ H}_k \leq E\}$ of the Toeplitz quantization $\hat{ H}_k$ of a smooth Hamiltonian $H: M \to \mathbb{ R}$. It is shown that the relative partial density of states $\frac{ \Pi_{ k, E}(z)}{ \Pi_k(z)} \to { 1}_{ \mathcal{ A}}$ where $\mathcal{ A} = \{ H < E\}$. Moreover it is shown that this partial density of states exhibits `Erf'-asymptotics along the interface $\partial \mathcal{ A}$, that is, the density profile asymptotically has a Gaussian error function shape interpolating between the values $1,0$ of ${ 1}_{ \mathcal{ A}}$. Such `erf'-asymptotics are a universal edge effect. The different types of scaling asymptotics are reminiscent of the law of large numbers and central limit theorem
Categories: math.CV math.SP