Decay for thewaveand schrodinger evolutions on manifolds with conical ends, part i

Wilhelm Schlag, Avy Soffer, Wolfgang Staubach

Research output: Contribution to journalArticle

15 Citations (Scopus)

Abstract

Let Omega; ? RN be a compact imbedded Riemannian manifold of dimension d > 1 and define the (d + l)-dimensional Riemannian manifold M := {(x,r(x)w): x G R,w? O} with r> 0 and smooth, and the natural metric ds2 = (1 + r'(x)2)dx2 + r2(x)ds2O.We require that M has conical ends: r(x) = |x| + 0(x-1)as x ? ±8. The Hamiltonian flow on such manifolds always exhibits trapping. Dispersive estimates for the Schrödinger evolution eit?M and the wave evolution eltv-?M are obtained for data of the form f (x, w)= Yn(W)u(x), where Yn are eigenfunctions of ?O. This paper treats the case d =1, Y0 = 1. In Part II of this paper we provide details for all cases d + n> 1. Our method combines two main ingredients: (A) A detailed scattering analysis of Schrodinger operators of the form -?2E + V(E) on the line where V(E) has inverse square behavior at infinity. (B) Estimation of oscillatory integrals by (non)stationary phase. © 2009 American Mathematical Society.

Original languageEnglish
Pages (from-to)19-52
Number of pages34
JournalTransactions of the American Mathematical Society
Volume362
Issue number1
DOIs
Publication statusPublished - Jan 2010

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