Scattering theory for symmetric spaces of noncompact type
type: Item
instance of: scholarly article
title: Scattering theory for symmetric spaces of noncompact type
zbMATH Open document ID: 0799.58081
DOI: 10.1215/S0012-7094-93-07201-8
author: Ralph S. Phillips
author: Mehrdad Shahshahani
published in: Duke Mathematical Journal
publication date: 1994-11-17T00:00:00Z
review text: Let \(G\) be a connected semisimple real Lie group with finite center and \(K\) a maximal compact subgroup. Let \(X = G/K\) be the associated symmetric space which is supposed to be of noncompact type. The authors study a hyperbolic system of equations, one for each time direction, whose solution is given by a commutative family of unitary operators on a suitable Hilbert space. The starting point is to expand the Fourier transform of the solution of the hyperbolic system in terms of the eigenvectors of the generators. Although the various components are no longer the Fourier transforms of data, they each provide a different translation representation, and their individual inverse Fourier transforms yield the solution of the original problem. Then different translation representations are used to study the associated outgoing subspaces. The \(\sigma\)-outgoing subspace \(D_ \sigma\) is defined as the set of all data whose \(\sigma\)-translation representers have their support in the dual of the positive Weyl chamber. \(D_{\sigma}\) satisfies a variant of the Lax-Phillips axioms. \(D^ 0_ \sigma\) is the space of all initial data to solutions which vanish in the \(\sigma\)-forward light cone. The main result of the paper asserts equality of the two spaces \(D_ \sigma = D^ 0_ \sigma\). The paper has its roots in the work of \textit{M. A. Semenov-Tyan-Shanskij} [Math. USSR, Izv. 10(1976), 535-563 (1977); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 40, 562-592 (1976; Zbl 0354.35071)].
Mathematics Subject Classification ID: 58J40
Mathematics Subject Classification ID: 35P25
zbMATH DE Number: 558992
zbMATH Keywords: hyperbolic equations
zbMATH Keywords: scattering theory
zbMATH Keywords: symmetric spaces
reviewed by: Jürgen Marschall
MaRDI profile type: Publication
cites work: A Basic Inequality for Scattering Theory on Riemannian Symmetric Spaces of the Noncompact Type
cites work: Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces
cites work: Spherical Functions on a Semisimple Lie Group, I
cites work: A duality for symmetric spaces with applications to group representations
cites work: The surjectivity of invariant differential operators on symmetric spaces. I
cites work: A duality for symmetric spaces with applications to group representations. II: Differential equations and eigenspace representations
cites work: Q3331943
cites work: Q3829998
cites work: On convexity, the Weyl group and the Iwasawa decomposition
cites work: Q4092184
cites work: Q5616408
cites work: Translation representations for the solution of the non-euclidean wave equation
cites work: Translation representations for the solution of the non-euclidean wave equation II
cites work: HARMONIC ANALYSIS ON RIEMANNIAN SYMMETRIC SPACES OF NEGATIVE CURVATURE AND SCATTERING THEORY
cites work: Q3665945
cites work: Poincare Inequality, Uncertainty Principle, and Scattering Theory on Symmetric Spaces
cites work: Q4751318
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description: scientific article; zbMATH DE number 558992