Central limit theorem for spectral partial Bergman kernels

摘要

Partial Bergman kernels Pi(k,E) are kernels of orthogonal projections onto subspaces S-k subset of H-0(M, L-k) of holomorphic sections of the kth power of an ample line bundle over a Kahler manifold (M, omega). The subspaces of this article are spectral subspaces {(H) over cap (k) <= E} of the Toeplitz quantization (H) over cap (k) of a smooth Hamiltonian H: M -> R. It is shown that the relative partial density of states satisfies Pi(k,E)(z)/Pi(k)(z) -> 1(A) where A = {H < E}. Moreover it is shown that this partial density of states exhibits "Erf" asymptotics along the interface partial derivative A; that is, the density profile asymptotically has a Gaussian error function shape interpolating between the values 1 and 0 of 1(A). Such "Erf" asymptotics are a universal edge effect. The different types of scaling asymptotics are reminiscent of the law of large numbers and the central limit theorem.