no. 5
The resolvent for Laplace-type operators on asymptotically conic spaces
[Résolvante des opérateurs du type laplacien sur les variétés asymptotiquement coniques]
Hassell, Andrew1 ; Vasy, András2
1 Australian National University, Centre for Mathematics and its Applications, Canberra ACT 0200 (Australie)
2 Massachusetts Institute of Technology, Department of Mathematics, Cambridge MA (USA)
Annales de l'Institut Fourier, Tome 51 (2001) no. 5, pp. 1299-1346.

Soient X une variété compacte à bord, et g une métrique de diffusion sur X qui est soit à courte portée, soit à longue portée du type gravitationnel. Alors (X,g) est une variété riemannienne complète asymptotiquement conique. Nous considérons l’opérateur H=Δ+P, où Δ est le laplacien de g et P est un opérateur différentiel de diffusion du premier ordre (formellement) auto-adjoint à coefficients s’annulant sur ∂X 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 H, R(σ+i0), pour σ>0. 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.

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=Δ+P, where Δ is the Laplacian with respect to g and P is a self-adjoint first order scattering differential operator with coefficients vanishing at ∂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(σ+i0), for σ 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.

DOI : 10.5802/aif.1856
Classification : 35P25, 58J40
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

1 Australian National University, Centre for Mathematics and its Applications, Canberra ACT 0200 (Australie)
2 Massachusetts Institute of Technology, Department of Mathematics, Cambridge MA (USA) 
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Hassell, Andrew; Vasy, András. The resolvent for Laplace-type operators on asymptotically conic spaces. Annales de l'Institut Fourier, Tome 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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