Eigenfunction expansions and scattering theory for the wave equation in an exterior region

1966 ◽  
Vol 21 (2) ◽  
pp. 120-150 ◽  
Author(s):  
Norman A. Shenk
Keyword(s):  
Wave Equation ◽  
Scattering Theory ◽  
Eigenfunction Expansions ◽  
Exterior Region

Spectral representation and scattering theory for the wave equation with two unbounded media

1993 ◽  
Vol 113 (2) ◽  
pp. 423-447 ◽  
Author(s):  
G. F. Roach ◽  
Bo Zhang
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Stationary Phase ◽  
Scattering Theory ◽  
Spectral Representation ◽  
Fourier Transforms ◽  
Eigenfunction Expansions ◽  
Wave Operators ◽  
Unbounded Media ◽  
Generalized Eigenfunction

AbstractIn this paper, we establish the generalized eigenfunction expansions for wave propagation in inhomogeneous, penetrable media in ℝn(n ≥ 2) with an unbounded interface. We then use them together with the method of stationary phase to prove the existence of the wave operators and to obtain the representations of the wave operators in terms of the generalized Fourier transforms.

scholarly journals A Non-degenerate Scattering Theory for the Wave Equation on Extremal Reissner–Nordström

2020 ◽  
Vol 380 (1) ◽  
pp. 323-408
Author(s):  
Yannis Angelopoulos ◽  
Stefanos Aretakis ◽  
Dejan Gajic
Keyword(s):  
Black Hole ◽  
Wave Equation ◽  
Event Horizon ◽  
Scattering Theory ◽  
Physical Space ◽  
Extremal Black Hole ◽  
Null Infinity ◽  
Interior Region ◽  
Exterior Region ◽  
Black Hole Interior

Abstract It is known that sub-extremal black hole backgrounds do not admit a (bijective) non-degenerate scattering theory in the exterior region due to the fact that the redshift effect at the event horizon acts as an unstable blueshift mechanism in the backwards direction in time. In the extremal case, however, the redshift effect degenerates and hence yields a much milder blueshift effect when viewed in the backwards direction. In this paper, we construct a definitive (bijective) non-degenerate scattering theory for the wave equation on extremal Reissner–Nordström backgrounds. We make use of physical-space energy norms which are non-degenerate both at the event horizon and at null infinity. As an application of our theory we present a construction of a large class of smooth, exponentially decaying modes. We also derive scattering results in the black hole interior region.

scholarly journals Scattering theory for the wave equation on a hyperbolic manifold

1987 ◽  
Vol 74 (2) ◽  
pp. 346-398 ◽  
Author(s):  
Ralph Phillips ◽  
Bettina Wiskott ◽  
Alex Woo
Keyword(s):  
Wave Equation ◽  
Scattering Theory ◽  
Hyperbolic Manifold

scholarly journals Scattering theory for the radialH˙12-critical wave equation with a cubic convolution

2015 ◽  
Vol 259 (12) ◽  
pp. 7199-7237 ◽  
Author(s):  
Changxing Miao ◽  
Junyong Zhang ◽  
Jiqiang Zheng
Keyword(s):  
Wave Equation ◽  
Scattering Theory

Scattering on Riemannian Symmetric Spaces and Huygens Principle

2018 ◽  
Vol 30 (08) ◽  
pp. 1840015
Author(s):  
Michael Semenov-Tian-Shansky
Keyword(s):  
Wave Equation ◽  
Scattering Theory ◽  
Symmetric Spaces ◽  
Evolution System ◽  
Semisimple Lie Groups ◽  
Curved Space ◽  
Zero Curvature ◽  
Famous Paper ◽  
Huygens Principle ◽  
Riemannian Symmetric Spaces

The famous paper by L. D. Faddeev and B. S. Pavlov (1972) on automorphic wave equation explored a highly romantic link between Scattering Theory (in the sense of Lax and Phillips) and Riemann hypothesis. An attempt to generalize this approach to general semisimple Lie groups leads to an interesting evolution system with multidimensional time explored by the author in 1976. In the present paper, we compare this system with a simpler one defined for zero curvature symmetric spaces and show that the Huygens principle for this system in the curved space holds if and only if it holds in the zero curvature limit.

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