Propagation of Singularities for Subelliptic Wave Equations

Hörmander’s propagation of singularities theorem does not fully describe the propagation of singularities in subelliptic wave equations, due to the existence of doubly characteristic points. In the present work, building upon a visionary conference paper by Melrose (in: Hyperbolic equations and rela...

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Bibliographic Details
Published in:Communications in mathematical physics 2022-10, Vol.395 (1), p.143-178
Main Author: Letrouit, Cyril
Format: Article
Language:English
Subjects:
Classical and Quantum Gravitation
Complex Systems
Control theory
Mathematical analysis
Mathematical and Computational Physics
Mathematical Physics
Optimal control
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Singularity (mathematics)
Theoretical
Wave equations
Wave propagation
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Summary:Hörmander’s propagation of singularities theorem does not fully describe the propagation of singularities in subelliptic wave equations, due to the existence of doubly characteristic points. In the present work, building upon a visionary conference paper by Melrose (in: Hyperbolic equations and related topics, Academic Press, pp 181–192, 1986), we prove that singularities of subelliptic wave equations only propagate along null-bicharacteristics and abnormal extremals, which are well-known curves in optimal control theory. As a consequence, we characterize the singular support of subelliptic wave kernels outside the diagonal. These results show that abnormal extremals play an important role in the classical-quantum correspondence between sub-Riemannian geometry and sub-Laplacians.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-022-04415-9