Diffractive Propagation on Conic Manifolds
Jared Wunsch1
1 Department of Mathematics Northwestern University 2033 Sheridan Rd. Evanston IL 60208 USA
Séminaire Laurent Schwartz — EDP et applications (2015-2016), Talk 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).

Published online:
DOI: 10.5802/slsedp.85
Jared Wunsch 1

1 Department of Mathematics Northwestern University 2033 Sheridan Rd. Evanston IL 60208 USA 
@article{SLSEDP_2015-2016____A9_0,
     author = {Jared Wunsch},
     title = {Diffractive {Propagation} on {Conic~Manifolds}},
     journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
     note = {talk:9},
     pages = {1--15},
     publisher = {Institut des hautes des scientifiques & Centre de mathtiques Laurent Schwartz, ole polytechnique},
     year = {2015-2016},
     doi = {10.5802/slsedp.85},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.85/}
}
TY  - JOUR
AU  - Jared Wunsch
TI  - Diffractive Propagation on Conic Manifolds
JO  - Séminaire Laurent Schwartz — EDP et applications
N1  - talk:9
PY  - 2015-2016
SP  - 1
EP  - 15
PB  - Institut des hautes des scientifiques & Centre de mathtiques Laurent Schwartz, ole polytechnique
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.85/
DO  - 10.5802/slsedp.85
LA  - en
ID  - SLSEDP_2015-2016____A9_0
ER  - 
%0 Journal Article
%A Jared Wunsch
%T Diffractive Propagation on Conic Manifolds
%J Séminaire Laurent Schwartz — EDP et applications
%Z talk:9
%D 2015-2016
%P 1-15
%I Institut des hautes des scientifiques & Centre de mathtiques Laurent Schwartz, ole polytechnique
%U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.85/
%R 10.5802/slsedp.85
%G en
%F SLSEDP_2015-2016____A9_0
Jared Wunsch. Diffractive Propagation on Conic Manifolds. Séminaire Laurent Schwartz — EDP et applications (2015-2016), Talk no. 9, 15 p. doi : 10.5802/slsedp.85. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.85/

[1] C. Bardos; J.-C. Guillot; J. Ralston 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] Dean Baskin; Jeremy L. Marzuola; Jared Wunsch 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] Dean Baskin; Jared Wunsch Resolvent estimates and local decay of waves on conic manifolds, J. Differential Geom., Volume 95 (2013) no. 2, pp. 183-214

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cited by Sources: