Diffractive Propagation on Conic Manifolds
Wunsch, Jared1
1 Department of Mathematics Northwestern University 2033 Sheridan Rd. Evanston IL 60208 USA
Séminaire Laurent Schwartz — EDP et applications (2015-2016), Exposé no. 9, 15 p.

In this survey, we review some applications and extensions of the author’s results with Richard Melrose on propagation of singularities for solutions to the wave equation on manifolds with conical singularities. These results mainly concern: the local decay of energy on noncompact manifolds with diffractive trapped orbits (joint work with Dean Baskin); singularities of the wave trace created by diffractive closed geodesics (joint work with G. Austin Ford); and the distribution of scattering resonances associated to such closed geodesics (joint work with Luc Hillairet).

Publié le :
DOI : 10.5802/slsedp.85
Wunsch, Jared 1

1 Department of Mathematics Northwestern University 2033 Sheridan Rd. Evanston IL 60208 USA 
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Wunsch, Jared. Diffractive Propagation on Conic Manifolds. Séminaire Laurent Schwartz — EDP et applications (2015-2016), Exposé no. 9, 15 p. doi : 10.5802/slsedp.85. http://www.numdam.org/articles/10.5802/slsedp.85/

[1] Bardos, C.; Guillot, J.-C.; Ralston, J. La relation de Poisson pour l’équation des ondes dans un ouvert non borné, Comm P.D.E., Volume 7 (1982), pp. 905-958

[2] Baskin, Dean; Marzuola, Jeremy L.; Wunsch, Jared Strichartz estimates on exterior polygonal domains, Geometric and spectral analysis (Contemp. Math.), Volume 630, Amer. Math. Soc., Providence, RI, 2014, pp. 291-306 | DOI

[3] Baskin, Dean; Wunsch, Jared Resolvent estimates and local decay of waves on conic manifolds, J. Differential Geom., Volume 95 (2013) no. 2, pp. 183-214

[4] Bérard, Pierre H On the wave equation on a compact Riemannian manifold without conjugate points, Mathematische Zeitschrift, Volume 155 (1977) no. 3, pp. 249-276

[5] Burq, Nicolas Pôles de diffusion engendrés par un coin, Astérisque (1997) no. 242, ii+122 pages

[6] Chandler-Wilde, Simon N; Graham, Ivan G; Langdon, Stephen; Spence, Euan A Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering, Acta numerica, Volume 21 (2012), pp. 89-305

[7] Chazarain, J. Formule de Poisson pour les variétés riemanniennes, Invent. Math., Volume 24 (1974), pp. 65-82

[8] Chazarain, Jacques Construction de la paramétrix du problème mixte hyperbolique pour l’équation des ondes, CR Acad. Sci. Paris, Volume 276 (1973), pp. 1213-1215

[9] Cheeger, J.; Taylor, M.E. On the diffraction of waves by conical singularities. I, Comm. Pure Appl. Math., Volume 35 (1982) no. 3, pp. 275-331

[10] Cheeger, J.; Taylor, M.E. On the diffraction of waves by conical singularities. II, Comm. Pure Appl. Math., Volume 35 (1982) no. 4, pp. 487-529

[11] Colin de Verdière, Yves Spectre du laplacien et longueurs des géodésiques périodiques. I, Compositio Mathematica, Volume 27 (1973) no. 1, pp. 83-106

[12] Colin de Verdière, Yves Spectre du laplacien et longueurs des géodésiques périodiques. II, Compositio Mathematica, Volume 27 (1973) no. 2, pp. 159-184

[13] Duistermaat, J.J.; Guillemin, V.W. The spectrum of positive elliptic operators and periodic geodesics, Invent. Math., Volume 29 (1975), pp. 39-79

[14] Duyckaerts, Thomas Inégalités de résolvante pour l’opérateur de Schrödinger avec potentiel multipolaire critique, Bull. Soc. Math. France, Volume 134 (2006) no. 2, pp. 201-239

[15] Ford, G. Austin; Wunsch, Jared The diffractive wave trace on manifolds with conic singularities (2014) (http://arxiv.org/abs/1411.6913)

[16] Galkowski, Jeffrey A quantitative Vainberg method for black box scattering (2015) (http://arxiv.org/abs/1511.05894)

[17] Hillairet, Luc Contribution of periodic diffractive geodesics, J. Funct. Anal., Volume 226 (2005) no. 1, pp. 48-89 | DOI

[18] Hormander, L Linear differential operators, Proc. internat. congress math. (Nice, 1970), Volume 1 (1963), pp. 121-133

[19] Lax, P.D.; Phillips, R.S. Scattering theory, Academic Press, New York, 1967 (Revised edition, 1989)

[20] Melrose, R.B. Microlocal parametrices for diffractive boundary value problems, Duke Math. J., Volume 42 (1975), pp. 605-635

[21] Melrose, R.B.; Sjöstrand, J. Singularities in boundary value problems I, Comm. Pure Appl. Math., Volume 31 (1978), pp. 593-617

[22] Melrose, R.B.; Sjöstrand, J. Singularities in boundary value problems II, Comm. Pure Appl. Math., Volume 35 (1982), pp. 129-168

[23] Melrose, Richard Scattering theory and the trace of the wave group, J. Funct. Anal., Volume 45 (1982) no. 1, pp. 29-40

[24] Melrose, Richard; Vasy, András; Wunsch, Jared Propagation of singularities for the wave equation on edge manifolds, Duke Math. J., Volume 144 (2008) no. 1, pp. 109-193

[25] Melrose, Richard; Vasy, András; Wunsch, Jared Diffraction of singularities for the wave equation on manifolds with corners, Astérisque (2013) no. 351, vi+135 pages

[26] Melrose, Richard; Wunsch, Jared Propagation of singularities for the wave equation on conic manifolds, Invent. Math., Volume 156 (2004) no. 2, pp. 235-299

[27] Morawetz, Cathleen S. Decay for solutions of the exterior problem for the wave equation, Comm. Pure Appl. Math., Volume 28 (1975), pp. 229-264

[28] Ralston, James V. Solutions of the wave equation with localized energy, Comm. Pure Appl. Math., Volume 22 (1969), pp. 807-823

[29] Sjöstrand, J.; Zworski, M. Lower bounds on the number of scattering poles, Comm. P.D.E., Volume 18 (1993), pp. 847-858

[30] Sjostrand, Johannes; Zworski, Maciej Lower bounds on the number of scattering poles, II, Journal of functional analysis, Volume 123 (1994) no. 2, pp. 336-367

[31] Sommerfeld, A. Mathematische theorie der diffraktion, Math. Annalen, Volume 47 (1896), pp. 317-374

[32] Tang, Siu-Hung; Zworski, Maciej Resonance expansions of scattered waves, Comm. Pure Appl. Math., Volume 53 (2000) no. 10, pp. 1305-1334

[33] Taylor, M.E. Grazing rays and reflection of singularities to wave equations, Comm. Pure Appl. Math., Volume 29 (1978), pp. 1-38

[34] Vaĭnberg, B. R. Asymptotic methods in equations of mathematical physics, Gordon & Breach Science Publishers, New York, 1989

[35] Vasy, András Propagation of singularities for the wave equation on manifolds with corners, Ann. of Math., Volume 168 (2008) no. 3, pp. 749-812 | DOI

[36] Wunsch, Jared A Poisson relation for conic manifolds, Math. Res. Lett., Volume 9 (2002) no. 5-6, pp. 813-828

[37] Zworski, Maciej Poisson formula for resonances in even dimensions (1999) (http://arxiv.org/abs/math/9901093)

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