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Hadamard expansions for powers of causal Green’s operators and “resolvents”

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The Hadamard expansion describes the singularity structure of Green’s operators associated with a normally hyperbolic operator P in terms of Riesz distributions (fundamental solutions on Minkowski space, transported to the manifold via the exponential map) and Hadamard coefficients (smooth sections in two variables, corresponding to the heat Kernel coefficients in the Riemannian case). In this paper, we derive an asymptotic expansion analogous to the Hadamard expansion for powers of advanced/retarded Green’s operators associated with P, as well as expansions for advanced/retarded Green’s operators associated with \(P-z\) for \(z\in \mathbb {C}\). These expansions involve the same Hadamard coefficients as the original Hadamard expansion, as well as the same or analogous (with built-in z-dependence) Riesz distributions.

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I would like to thank Matthias Lesch and Koen van den Dungen for their advice and support during the writing of both the thesis that this paper is based on and the paper itself. I also thank the anonymous referee for reviewing my paper and pointing out mistakes that I had overlooked.


During the writing of this paper, the author was employed at the University of Bonn. No further funding was received.

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Correspondence to Lennart Ronge.

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Ronge, L. Hadamard expansions for powers of causal Green’s operators and “resolvents”. Ann Glob Anal Geom 64, 16 (2023).

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