TY - JOUR
T1 - Decay for thewaveand schrodinger evolutions on manifolds with conical ends, part i
AU - Schlag, Wilhelm
AU - Soffer, Avy
AU - Staubach, Wolfgang
PY - 2010/1
Y1 - 2010/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=77950897210&partnerID=8YFLogxK
U2 - 10.1090/S0002-9947-09-04690-X
DO - 10.1090/S0002-9947-09-04690-X
M3 - Article
SN - 0002-9947
VL - 362
SP - 19
EP - 52
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
IS - 1
ER -