Let be a compact manifold with boundary, and a scattering metric on , which may be either of short range or “gravitational” long range type. Thus, gives the geometric structure of a complete manifold with an asymptotically conic end. Let be an operator of the form , where is the Laplacian with respect to and is a self-adjoint first order scattering differential operator with coefficients vanishing at 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 , , for 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.
Soient une variété compacte à bord, et une métrique de diffusion sur qui est soit à courte portée, soit à longue portée du type gravitationnel. Alors est une variété riemannienne complète asymptotiquement conique. Nous considérons l’opérateur , où est le laplacien de et est un opérateur différentiel de diffusion du premier ordre (formellement) auto-adjoint à coefficients s’annulant sur et satisfaisant une condition gravitationnelle. Nous définissons un calcul symbolique pour les distributions de Legendre sur les variétés compactes à coins de codimension deux, et nous l’utilisons pour une construction directe du noyau de la résolvante de , , pour . Cette approche n’utilise pas le principe d’absorption limite. Au lieu de cela nous construisons une paramétrixe qui satisfait l’équation de la résolvante à un terme d’erreur compacte près qui est éliminé grâce à la théorie de Fredholm.
Keywords: Legendre distributions, symbol calculus, scattering metrics, resolvent kernel
Mot clés : distributions de Legendre, calcul symbolique, métriques de diffusion, noyau résolvant
Hassell, Andrew 1; Vasy, András 2
@article{AIF_2001__51_5_1299_0, author = {Hassell, Andrew and Vasy, Andr\'as}, title = {The resolvent for {Laplace-type} operators on asymptotically conic spaces}, journal = {Annales de l'Institut Fourier}, pages = {1299--1346}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {51}, number = {5}, year = {2001}, doi = {10.5802/aif.1856}, zbl = {0983.35098}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1856/} }
TY - JOUR AU - Hassell, Andrew AU - Vasy, András TI - The resolvent for Laplace-type operators on asymptotically conic spaces JO - Annales de l'Institut Fourier PY - 2001 SP - 1299 EP - 1346 VL - 51 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1856/ DO - 10.5802/aif.1856 LA - en ID - AIF_2001__51_5_1299_0 ER -
%0 Journal Article %A Hassell, Andrew %A Vasy, András %T The resolvent for Laplace-type operators on asymptotically conic spaces %J Annales de l'Institut Fourier %D 2001 %P 1299-1346 %V 51 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1856/ %R 10.5802/aif.1856 %G en %F AIF_2001__51_5_1299_0
Hassell, Andrew; Vasy, András. The resolvent for Laplace-type operators on asymptotically conic spaces. Annales de l'Institut Fourier, Volume 51 (2001) no. 5, pp. 1299-1346. doi : 10.5802/aif.1856. https://aif.centre-mersenne.org/articles/10.5802/aif.1856/
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