Propagation of singularities for the wave equation on manifolds with corners

Vasy, András

doi:10.4007/annals.2008.168.749

Cited by 69 publications

(224 citation statements)

References 18 publications

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“…This fact follows from work of A. Vasy [Vas05;Vas08] on the Poisson relation for manifolds with corners. In other words, the presence of corners does not affect the wave trace expansion at the bouncing ball orbit.…”

Section: Statement Of Resultsmentioning

confidence: 67%

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Inverse spectral problem for analytic domains, II: ℤ₂-symmetric domains

Zelditch

2009

Ann. Math.

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This paper develops and implements a new algorithm for calculating wave trace invariants of a bounded plane domain around a periodic billiard orbit. The algorithm is based on a new expression for the localized wave trace as a special multiple oscillatory integral over the boundary, and on a Feynman diagrammatic analysis of the stationary phase expansion of the oscillatory integral. The algorithm is particularly effective for Euclidean plane domains possessing a ‫ޚ‬ 2 symmetry which reverses the orientation of a bouncing ball orbit. It is also very effective for domains with dihedral symmetries. For simply connected analytic Euclidean plane domains in either symmetry class, we prove that the domain is determined within the class by either its Dirichlet or Neumann spectrum. This improves and generalizes the best prior inverse result that simply connected analytic plane domains with two symmetries are spectrally determined within that class.

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Section: Statement Of Resultsmentioning

confidence: 67%

“…1. A. Vasy and J. Wunsch gave advice on [Vas05]. We thank Y. Colin de Verdière and attendees of the spectral theory seminar at Grenoble for their patience in listening to earlier versions of this article and for their comments.…”

Section: Acknowledgmentsmentioning

confidence: 99%

Inverse spectral problem for analytic domains, II: ℤ₂-symmetric domains

Zelditch

2009

Ann. Math.

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“…27. The diffractive improvement in a model case, namely edge manifolds, defined in the last section, is proved in Ref.…”

Section: Introductionmentioning

confidence: 86%

Diffraction by Edges

Vasy

2008

Mod. Phys. Lett. B

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In these expository notes we explain the role of geometric optics in wave propagation on domains or manifolds with corners or edges. Both the propagation of singularities, which describes where solutions of the wave equation may be singular, and the diffractive improvement under non-focusing hypotheses, which states that in certain places the diffracted wave is more regular than a priori expected, is described. In addition, the wave equation on differential forms with natural boundary conditions, which in particular includes a formulation of Maxwell's equations, is studied.

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“…We suggest that the reader compares these statements to (2.1) and the surrounding discussion. There is also a corresponding space of pseudo-differential operators, Ψ m b (X), introduced by Melrose in [25]; see [27] for a thorough description and [39] for a summary. Elements of Ψ m b (X) have principal symbols…”

Section: Conormal Regularitymentioning

confidence: 99%

The wave equation on asymptotically de Sitter-like spaces

Vasy

2010

Advances in Mathematics

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145

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In this paper we obtain the asymptotic behavior of solutions of the Klein-Gordon equation on Lorentzian manifolds (X • , g) which are de Sitter-like at infinity. Such manifolds are Lorentzian analogues of the socalled Riemannian conformally compact (or asymptotically hyperbolic) spaces. Under global assumptions on the (null)bicharacteristic flow, namely that the boundary of the compactification X is a union of two disjoint manifolds, Y ± , and each bicharacteristic converges to one of these two manifolds as the parameter along the bicharacteristic goes to +∞, and to the other manifold as the parameter goes to −∞, we also define the scattering operator, and show that it is a Fourier integral operator associated to the bicharacteristic flow from Y + to Y − .

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Propagation of singularities for the wave equation on manifolds with corners

Cited by 69 publications

References 18 publications

Inverse spectral problem for analytic domains, II: ℤ₂-symmetric domains

Inverse spectral problem for analytic domains, II: ℤ₂-symmetric domains

Diffraction by Edges

The wave equation on asymptotically de Sitter-like spaces

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Propagation of singularities for the wave equation on manifolds with corners

Cited by 69 publications

References 18 publications

Inverse spectral problem for analytic domains, II: ℤ2-symmetric domains

Inverse spectral problem for analytic domains, II: ℤ2-symmetric domains

Diffraction by Edges

The wave equation on asymptotically de Sitter-like spaces

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Inverse spectral problem for analytic domains, II: ℤ₂-symmetric domains

Inverse spectral problem for analytic domains, II: ℤ₂-symmetric domains