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 Colin Guillarmou; Andrew HassellResolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. II(Résolvante à basse énergie et transformée de Riesz pour l’opérateur de Schrödinger sur des variétés asymptotiquement coniques. II)Annales de l'institut Fourier, 59 no. 4 (2009), p. 1553-1610, doi: 10.5802/aif.2471 Article PDF | Reviews MR 2566968 | Zbl 1175.58011 Class. Math.: 58J50, 42B20, 35J10Keywords: Asymptotically conic manifold, scattering metric, resolvent kernel, low energy asymptotics, Riesz transform, zero-resonance Résumé - AbstractLet $M^\circ$ be a complete noncompact manifold of dimension at least 3 and $g$ an asymptotically conic metric on $M^\circ$, in the sense that $M^\circ$ compactifies to a manifold with boundary $M$ so that $g$ becomes a scattering metric on $M$. We study the resolvent kernel $(P + k^2)^{-1}$ and Riesz transform $T$ of the operator $P = \Delta _g + V$, where $\Delta _g$ is the positive Laplacian associated to $g$ and $V$ is a real potential function smooth on $M$ and vanishing at the boundary. In our first paper we assumed that $P$ has neither zero modes nor a zero-resonance and showed (i) that the resolvent kernel is polyhomogeneous conormal on a blown up version of $M^2 \times [0, k_0]$, and (ii) $T$ is bounded on $L^p(M^\circ )$ for $1 < p < n$, which range is sharp unless $V \equiv 0$ and $M^\circ$ 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 $n=4$ and there is a zero-resonance) the resolvent kernel is polyhomogeneous on the same space, and we find the precise range of $p$ (generically $n/(n-2) < p < n/3$) for which $T$ is bounded on $L^p(M)$ when zero modes are present. Bibliography[1] M. Abramowitz & I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series 55, Dover Publications, 1964  MR 167642 |  Zbl 0643.33001[2] S. Agmon, Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of $N$-body Schrödinger operators, Mathematical Notes 29, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1982  MR 745286 |  Zbl 0503.35001[3] G. Carron, “A topological criterion for the existence of half-bound states”, J. London Math. Soc. 65 (2002), p. 757-768 Article |  MR 1895746 |  Zbl 1027.58023[4] G. Carron, T. Coulhon & A. Hassell, “Riesz transform and $L^p$ cohomology for manifolds with Euclidean ends”, Duke Math. J. 133 (2006) no. 1, p. 59-93 Article |  MR 2219270 |  Zbl 1106.58021[5] C. Guillarmou & A. Hassell, “Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. I.”, Math. Ann. 341 (2008) no. 4, p. 859-896 Article |  MR 2407330 |  Zbl 1141.58017[6] A. Jensen, “Spectral properties of Schrödinger operators and time-decay of the wave functions: results in $L^2(\mathbb{R}^m), m \ge 5$”, Duke Math. J. 47 (1980), p. 57-80 Article |  MR 563367 |  Zbl 0437.47009[7] A. Jensen & T. Kato, “Spectral properties of Schrödinger operators and time-decay of the wave functions”, Duke Math. J. 46 (1979), p. 583-611 Article |  MR 544248 |  Zbl 0448.35080[8] H.-Q. Li, “La transformée de Riesz sur les variétés coniques”, J. Funct. Anal. 168 (1999) no. 1, p. 145-238 Article |  MR 1717835 |  Zbl 0937.43004[9] R. B. Melrose, “Calculus of conormal distributions on manifolds with corners”, Int. Math. Res. Not. 3 (1992), p. 51-61 Article |  MR 1154213 |  Zbl 0754.58035[10] R. B. Melrose, The Atiyah-Patodi-Singer index theorem, AK Peters, Wellesley, 1993  MR 1348401 |  Zbl 0796.58050[11] M. Murata, “Asymptotic expansions in time for solutions of Schrödinger-Type Equations”, 49 (1982), p. 10-56  MR 680855 |  Zbl 0499.35019[12] X-P. Wang, “Asymptotic expansion in time of the Schrödinger group on conical manifolds”, to appear, Annales Inst. Fourier 2006 Cedram |  MR 2282678 |  Zbl 1118.35022 © Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310