Let be a complete noncompact manifold of dimension at least 3 and an asymptotically conic metric on , in the sense that compactifies to a manifold with boundary so that becomes a scattering metric on . We study the resolvent kernel and Riesz transform of the operator , where is the positive Laplacian associated to and is a real potential function smooth on and vanishing at the boundary.
In our first paper we assumed that has neither zero modes nor a zero-resonance and showed (i) that the resolvent kernel is polyhomogeneous conormal on a blown up version of , and (ii) is bounded on for , which range is sharp unless and has only one end.
In the present paper, we perform a similar analysis allowing zero modes and zero-resonances. We show once again that (unless and there is a zero-resonance) the resolvent kernel is polyhomogeneous on the same space, and we find the precise range of (generically ) for which is bounded on when zero modes are present.
Soit une variété complète de dimension et une métrique asymptotiquement conique sur , au sens où se compactifie en une variété à bord telle que soit une métrique de type “scattering” sur . On étudie le noyau intégral de la résolvante et la transformée de Riesz de l’opérateur , où est le laplacien positif associé à et un potentiel réel, lisse sur et s’annulant au bord.
Dans le premier article nous avons supposé que n’est ni résonance ni valeur propre pour et montré (i) que le noyau de la résolvante est conormal polyhomogène sur une version éclatée de , et (ii) que est borné sur pour , ce qui optimal sauf si ou bien a seulement un bout.
Dans le présent article, on effectue une analyse similaire tout en autorisant les cas où est résonance ou valeur propre. On montre là encore (sauf si et est résonance) que le noyau de la résolvante est polyhomogène sur le même espace, et on donne l’intervalle de (génériquement ) pour lequel est borné sur quand est valeur propre.
Keywords: Asymptotically conic manifold, scattering metric, resolvent kernel, low energy asymptotics, Riesz transform, zero-resonance
Mot clés : variété asymptotiquement conique, métrique scattering, noyau de la résolvante, asymptotique à basse énergie, transformée de Riesz, zéro-résonance
@article{AIF_2009__59_4_1553_0, author = {Guillarmou, Colin and Hassell, Andrew}, title = {Resolvent at low energy and {Riesz} transform for {Schr\"odinger} operators on asymptotically conic manifolds. {II}}, journal = {Annales de l'Institut Fourier}, pages = {1553--1610}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {59}, number = {4}, year = {2009}, doi = {10.5802/aif.2471}, zbl = {1175.58011}, mrnumber = {2566968}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2471/} }
TY - JOUR AU - Guillarmou, Colin AU - Hassell, Andrew TI - Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. II JO - Annales de l'Institut Fourier PY - 2009 SP - 1553 EP - 1610 VL - 59 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2471/ DO - 10.5802/aif.2471 LA - en ID - AIF_2009__59_4_1553_0 ER -
%0 Journal Article %A Guillarmou, Colin %A Hassell, Andrew %T Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. II %J Annales de l'Institut Fourier %D 2009 %P 1553-1610 %V 59 %N 4 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2471/ %R 10.5802/aif.2471 %G en %F AIF_2009__59_4_1553_0
Guillarmou, Colin; Hassell, Andrew. Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. II. Annales de l'Institut Fourier, Volume 59 (2009) no. 4, pp. 1553-1610. doi : 10.5802/aif.2471. http://archive.numdam.org/articles/10.5802/aif.2471/
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