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The propagation of singularities along gliding rays

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LetP be a second-order differential operator with real principal symbol and fibre-simple characteristics on a manifold with boundary non-characteristic forP. LetB be a differential operator such that the boundary value problem (P, B) is normal and satisfies the Lopatinskii-Schapiro condition. The singularities of distributions,u, such thatP u is smooth on the boundary, near points at which the boundary is bicharacteristically convex are shown to propagate, in the boundary, only along the gliding rays, which are the leaves of the Hamilton foliation of the glancing surface. This analysis, combined with known results on diffraction, leads to a Poisson relation bounding the singular support of the Fourier transform of the Dirichlet spectral density for a compact Riemannian manifold with geodesically convex, or concave, boundary in terms of the geodesic length spectrum.

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Authors and Affiliations

  1. Department of Mathematics, University of Lund, S 22007, Lund 7

    K. G. Andersson

  2. Department of mathematics, MIT, 02139, Cambridge, Mass., USA

    R. B. Melrose

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  1. K. G. Andersson
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  2. R. B. Melrose
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Andersson, K.G., Melrose, R.B. The propagation of singularities along gliding rays. Invent Math 41, 197–232 (1977). https://doi.org/10.1007/BF01403048

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