With cedram.org English français Home Overview Search for an article Submit a paper Informations for the authors Subscription Limited access - RUCHE Table of contents for this issue | Previous article | Next article Andrew Hassell; András VasyThe resolvent for Laplace-type operators on asymptotically conic spaces(Résolvante des opérateurs du type laplacien sur les variétés asymptotiquement coniques)Annales de l'institut Fourier, 51 no. 5 (2001), p. 1299-1346, doi: 10.5802/aif.1856 Article PDF | Reviews Zbl 0983.35098 Class. Math.: 35P25, 58J40Keywords: Legendre distributions, symbol calculus, scattering metrics, resolvent kernel Résumé - Abstract Let $X$ be a compact manifold with boundary, and $g$ a scattering metric on $X$, which may be either of short range or "gravitational" long range type. Thus, $g$ gives $X$ the geometric structure of a complete manifold with an asymptotically conic end. Let $H$ be an operator of the form $H =\Delta + P$, where $\Delta$ is the Laplacian with respect to $g$ and $P$ is a self-adjoint first order scattering differential operator with coefficients vanishing at $\partial X$ and satisfying a "gravitational" condition. We define a symbol calculus for Legendre distributions on manifolds with codimension two corners and use it to give a direct construction of the resolvent kernel of $H$, $R(\sigma + i0)$, for $\sigma$ on the positive real axis. In this approach, we do not use the limiting absorption principle at any stage; instead we construct a parametrix which solves the resolvent equation up to a compact error term and then use Fredholm theory to remove the error term. Bibliography[1] C. Gérard, H. Isozaki & E. Skibsted, “Commutator algebra and resolvent estimates”, Advanced Studies in Pure Mathematics vol. 23 (1994)  MR 1275395 |  Zbl 0814.35086[2] A. Hassell, “Distorted plane waves for the 3 body Schrödinger operator”, Geom. Funct. Anal. 10 (2000), p. 1-50 Article |  MR 1748915 |  Zbl 0953.35122[3] A. Hassell & A. Vasy, “Symbolic functional calculus and N-body resolvent estimates”, J. Funct. Anal. 173 (2000), p. 257-283 Article |  MR 1760615 |  Zbl 0960.58025[4] A. Hassell & A. Vasy, “The spectral projections and the resolvent for scattering metrics”, J. d'Anal. Math. 79 (1999), p. 241-298 Article |  MR 1749314 |  Zbl 0981.58025[5] A. Hassell & A. Vasy, “Legendrian distributions on manifolds with corners”, In preparation [6] L. Hörmander, “Fourier integral operators, I”, Acta Mathematica 127 (1971), p. 79-183 Article |  MR 388463 |  Zbl 0212.46601[7] L. Hörmander, The analysis of linear partial differential operators, III, Springer, 1983  MR 781536 |  Zbl 0601.35001[8] R. B. Melrose, “Calculus of conormal distributions on manifolds with corners”, International Mathematics Research Notices (1992) no. 3, p. 51-61 Article |  MR 1154213 |  Zbl 0754.58035[9] R. B. Melrose, The Atiyah-Patodi-Singer index theorem, A. K. Peters, Wellesley, MA, 1993  MR 1348401 |  Zbl 0796.58050[10] R. B. Melrose, Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces, Marcel Dekker, 1994  MR 1291640 |  Zbl 0837.35107[11] R. B. Melrose & G. Uhlmann, “Lagrangian Intersection and the Cauchy problem”, Comm. Pure and Appl. Math. 32 (1979), p. 483-519 Article |  MR 528633 |  Zbl 0396.58006[12] R. B. Melrose & M. Zworski, “Scattering metrics and geodesic flow at infinity”, Inventiones Mathematicae 124 (1996), p. 389-436 Article |  MR 1369423 |  Zbl 0855.58058[13] A. Vasy, “Geometric scattering theory for long-range potentials and metrics”, Int. Math. Res. Notices (1998), p. 285-315 Article |  MR 1616722 |  Zbl 0922.58085[14] J. Wunsch & M. Zworski, “Distribution of resonances for asymptotically euclidean manifolds”, To appear in J. Diff. Geom. Article |  MR 1849026 |  Zbl 1030.58024 © Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310