In a recent series of articles, the authors have studied the transition behavior of partial Bergman kernels Πk,[E1,E2 ](z, w) and the associated DOS (density of states) Πk,[E1,E2 ](z) across the interface C between the allowed and forbidden regions. Partial Bergman kernels are Toeplitz Hamiltonians quantizing Morse functions H:M →ℝ on a Kähler manifold. The allowed region is H−1([E1,E2 ]) and the interface C is its boundary. In prior articles it was assumed that the endpoints Ej were regular values of H. This article completes the series by giving parallel results when an endpoint is a critical value of H. In place of the Erf scaling asymptotics in a k−12 tube around C for regular interfaces, one obtains δ-asymptotics in k−14-tubes around singular points of a critical interface. In k−12 tubes, the transition law is given by the osculating metaplectic propagator.
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