The resolvent for Laplace-type operators on asymptotically conic spaces
Annales de l'Institut Fourier, Volume 51 (2001) no. 5, p. 1299-1346
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.
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.
DOI : https://doi.org/10.5802/aif.1856
Classification:  35P25,  58J40
Keywords: Legendre distributions, symbol calculus, scattering metrics, resolvent kernel
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {51},
     number = {5},
     year = {2001},
     pages = {1299-1346},
     doi = {10.5802/aif.1856},
     zbl = {0983.35098},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/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/item/AIF_2001__51_5_1299_0/

[1] C. Gérard; H. Isozaki; E. Skibsted Commutator algebra and resolvent estimates, Advanced Studies in Pure Mathematics, Tome vol. 23 (1994) | MR 1275395 | Zbl 0814.35086

[2] A. Hassell Distorted plane waves for the 3 body Schrödinger operator, Geom. Funct. Anal., Tome 10 (2000), pp. 1-50 | Article | MR 1748915 | Zbl 0953.35122

[3] A. Hassell; A. Vasy Symbolic functional calculus and N-body resolvent estimates, J. Funct. Anal., Tome 173 (2000), pp. 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., Tome 79 (1999), pp. 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, Tome 127 (1971), pp. 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, pp. 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., Tome 32 (1979), pp. 483-519 | Article | MR 528633 | Zbl 0396.58006

[12] R. B. Melrose; M. Zworski Scattering metrics and geodesic flow at infinity, Inventiones Mathematicae, Tome 124 (1996), pp. 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), pp. 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.) | MR 1849026 | Zbl 1030.58024