Partial Bergman kernels Π k,E are kernels of orthogonal projections onto subspaces S k ⊂ H 0 (M, L k ) of holomorphic sections of the kth power of an ample line bundle over a Kähler manifold (M, ω). The subspaces of this article are spectral subspaces {Ĥ k ≤ E} of the Toeplitz quantizationĤ k of a smooth Hamiltonian H : M → R. It is shown that the relative partial density of statesMoreover it is shown that this partial density of states exhibits 'Erf'-asymptotics along the interface ∂A, that is, the density profile asymptotically has a Gaussian error function shape interpolating between the values 1, 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 central limit theorem.

We prove a conjecture of the first author relating the Bernstein-Sato ideal of a finite collection of multivariate polynomials with cohomology support loci of rank one complex local systems. This generalizes a classical theorem of Malgrange and Kashiwara relating the b-function of a multivariate polynomial with the monodromy eigenvalues on the Milnor fibers cohomology. Contents 1. Introduction 1 2. The support of the specialization complex 3 3. Relative holonomic modules 6 4. Appendix 17 References 19

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