Scattering theory for symmetric spaces of noncompact type
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)].
scientific article; zbMATH DE number 558992
scientific article