Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. II.

*(English)*Zbl 1175.58011Summary: Let \(M^\circ\) 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 [Math. Ann. 341, No. 4, 859–896 (2008; Zbl 1141.58017)], 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.

In our first paper [Math. Ann. 341, No. 4, 859–896 (2008; Zbl 1141.58017)], 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.

##### MSC:

58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |

42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |

35J10 | Schrödinger operator, Schrödinger equation |

47F05 | General theory of partial differential operators |

##### Keywords:

asymptotically conic manifold; scattering metric; resolvent kernel; low energy asymptotics; Riesz transform; zero-resonance
PDF
BibTeX
XML
Cite

\textit{C. Guillarmou} and \textit{A. Hassell}, Ann. Inst. Fourier 59, No. 4, 1553--1610 (2009; Zbl 1175.58011)

**OpenURL**

##### References:

[1] | Abramowitz, M.; Stegun, I. A., Handbook of mathematical functions with formulas, graphs, and mathematical tables, 55, (1964), Dover Publications · Zbl 0643.33001 |

[2] | Agmon, S., Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of \(N\)-body Schrödinger operators, 29, (1982), Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo · Zbl 0503.35001 |

[3] | Carron, G., A topological criterion for the existence of half-bound states, J. London Math. Soc., 65, 757-768, (2002) · Zbl 1027.58023 |

[4] | Carron, G.; Coulhon, T.; Hassell, A., Riesz transform and \(L^p\) cohomology for manifolds with Euclidean ends, Duke Math. J., 133, 1, 59-93, (2006) · Zbl 1106.58021 |

[5] | Guillarmou, C.; Hassell, A., Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. I., Math. Ann., 341, 4, 859-896, (2008) · Zbl 1141.58017 |

[6] | Jensen, A., 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, 57-80, (1980) · Zbl 0437.47009 |

[7] | Jensen, A.; Kato, T., Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J., 46, 583-611, (1979) · Zbl 0448.35080 |

[8] | Li, H.-Q., La transformée de Riesz sur LES variétés coniques, J. Funct. Anal., 168, 1, 145-238, (1999) · Zbl 0937.43004 |

[9] | Melrose, R. B., Calculus of conormal distributions on manifolds with corners, Int. Math. Res. Not., 3, 51-61, (1992) · Zbl 0754.58035 |

[10] | Melrose, R. B., The Atiyah-Patodi-Singer index theorem, (1993), AK Peters, Wellesley · Zbl 0796.58050 |

[11] | Murata, M., Asymptotic expansions in time for solutions of Schrödinger-type equations, 49, 10-56, (1982) · Zbl 0499.35019 |

[12] | Wang, X-P., Asymptotic expansion in time of the Schrödinger group on conical manifolds, to appear, Annales Inst. Fourier, (2006) · Zbl 1118.35022 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.