The Hadamard parametrix on globally hyperbolic spacetimes with Robin boundary conditions: Fundamental solutions and Hadamard states
Abstract
On a d-dimensional, d≥ 2, globally hyperbolic spacetime with a timelike boundary (ℳ,g), we investigate the Klein-Gordon equation with Robin boundary conditions. First of all, we prove existence and uniqueness of the advanced and retarded fundamental solutions, discussing in addition their structural properties. Secondly, under the hypothesis of infinitesimal convexity of ∂ℳ and of local finiteness of boundary reflections for broken null bicharacteristics, we characterize their wavefront set in the interior mathringℳ, using the propagation of singularities theorem of Melrose and Sjöstrand. In the second part of the paper, we introduce a notion of Robin-Hadamard two-point correlation function as a positive distribution on ℳ×ℳ with prescribed antisymmetric part and singular structure, the latter being characterized in terms of its wavefront set. We complement this definition with a local Hadamard form. Up to a smooth remainder, this consists of the standard directed Hadamard parametrix together with additional contributions associated with the reflected null rays. Their coefficients are determined by transport equations supplemented by Robin matching conditions at the boundary. The construction applies to an arbitrary, but finite number of reflections. We prove that this local formulation is equivalent to the wavefront set condition, thereby establishing a counterpart of Radzikowski's theorem in the presence of a timelike boundary. Finally, starting from the ground state on suitable static backgrounds and using a deformation argument, we prove existence of Robin-Hadamard two-point correlation functions on the class of spacetimes considered.
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BibTeX
@misc{costeri2026hadamard,
title = {{The Hadamard parametrix on globally hyperbolic spacetimes with Robin boundary conditions: Fundamental solutions and Hadamard states}},
author = {Beatrice Costeri and Claudio Dappiaggi and Benito Alberto Juárez-Aubry},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24465},
eprint = {2609.24465},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24465},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Hadamard parametrix on globally hyperbolic spacetimes with Robin boundary conditions: Fundamental solutions and Hadamard states
AU - Beatrice Costeri
AU - Claudio Dappiaggi
AU - Benito Alberto Juárez-Aubry
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24465
UR - https://arxiv.org/abs/2609.24465
AB - On a d-dimensional, d≥ 2, globally hyperbolic spacetime with a timelike boundary (ℳ,g), we investigate the Klein-Gordon equation with Robin boundary conditions. First of all, we prove existence and uniqueness of the advanced and retarded fundamental solutions, discussing in addition their structural properties. Secondly, under the hypothesis of infinitesimal convexity of ∂ℳ and of local finiteness of boundary reflections for broken null bicharacteristics, we characterize their wavefront set in the interior mathringℳ, using the propagation of singularities theorem of Melrose and Sjöstrand. In the second part of the paper, we introduce a notion of Robin-Hadamard two-point correlation function as a positive distribution on ℳ×ℳ with prescribed antisymmetric part and singular structure, the latter being characterized in terms of its wavefront set. We complement this definition with a local Hadamard form. Up to a smooth remainder, this consists of the standard directed Hadamard parametrix together with additional contributions associated with the reflected null rays. Their coefficients are determined by transport equations supplemented by Robin matching conditions at the boundary. The construction applies to an arbitrary, but finite number of reflections. We prove that this local formulation is equivalent to the wavefront set condition, thereby establishing a counterpart of Radzikowski's theorem in the presence of a timelike boundary. Finally, starting from the ground state on suitable static backgrounds and using a deformation argument, we prove existence of Robin-Hadamard two-point correlation functions on the class of spacetimes considered.
ER -