Mathematische Annalen

, Volume 341, Issue 4, pp 859–896

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

Article

Abstract

Let \(M^\circ\) be a complete noncompact manifold and g an asymptotically conic manifold on \(M^\circ\), in the sense that \(M^\circ\) compactifies to a manifold with boundary M in such a way that g becomes a scattering metric on M. A special case that we focus on is that of asymptotically Euclidean manifolds, where the induced metric at infinity is equal to the standard metric on Sn−1; such manifolds have an end that can be identified with \({\mathbb{R}}^n \backslash B(R,0)\) in such a way that the metric is asymptotic in a precise sense to the flat Euclidean metric. We analyze the asymptotics of the resolvent kernel (P + k2)−1 where \(P = \Delta_g + V\) is the sum of the positive Laplacian associated to g and a real potential function \(V\in C^{\infty}(M)\) which vanishes to second order at the boundary (i.e. decays to second order at infinity on \(M^\circ\)) and such that \(\Delta_{\partial M}+(n-2)^2/4+V_0 > 0\) if \(V_0:=(x^{-2}V)|_{\partial M}\) . Then we show that on a blown up version of \(M^2 \times [0, k_0]\) the resolvent kernel is conormal to the lifted diagonal and polyhomogeneous at the boundary, and we are able to identify explicitly the leading behaviour of the kernel at each boundary hypersurface. Using this we show that the Riesz transform of P is bounded on \(L^p(M^\circ)\) for 1 < p < n if \(V_0\equiv 0\) , and that this range is optimal if \(V \not\equiv 0\) or if M has more than one end. The result with \(V\not\equiv 0\) is new even when \(M^\circ = {\mathbb{R}}^n\) , g is the Euclidean metric and V is compactly supported. When V ≡ 0 with one end, the range of p becomes 1 <  ppmax where pmax > n depends explicitly on the first non-zero eigenvalue of the Laplacian on the boundary \(\partial M\) . Our results hold for all dimensions ≥ 3 under the assumption that P has neither zero modes nor a zero-resonance. In the follow-up paper Guillarmou and Hassell (Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds, preprint) [7] we analyze the same situation in the presence of zero modes and zero-resonances.

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References

  1. 1.
    Agmon, S.: 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/University of Tokyo Press, Princeton/Tokyo (1982)Google Scholar
  2. 2.
    Auscher, P., Ben Ali, B.: Maximal Inequalities and Riesz Transform Estimates on L p Spaces for Schrödinger Operators with Nonnegative Potentials, Arxiv math.AP/0605047 (preprint)Google Scholar
  3. 3.
    Auscher P., Coulhon T., Duong X.T. and Hoffman S. (2004). Riesz transform on manifolds and heat kernel regularity. Ann. E.N.S. 37: 911–957 MATHGoogle Scholar
  4. 4.
    Carron G., Coulhon T. and Hassell A. (2006). Riesz transform and L p cohomology for manifolds with Euclidean ends. Duke Math. J. 133(1): 59–93 MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Coulhon, T., Dungey, N.: Riesz transform and perturbation. J. Geom. Anal. (to appear)Google Scholar
  6. 6.
    Coulhon T. and Duong X.T. (1999). Riesz transform for 1 ≤ p ≤ 2. Trans. A.M.S. 351: 1151–1169 MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Guillarmou, C., Hassell, A.: Resolvent at Low Energy and Riesz Transform for Schrödinger Operators on Asymptotically Conic Manifolds. II, arXiv:math/0703316 (preprint)Google Scholar
  8. 8.
    Hassell, A., Marshall, S.: Eigenvalues of Schrödinger operators with potential asymptotically homogeneous of degree −2. Trans. Am. Math. Soc. (to appear)Google Scholar
  9. 9.
    Hassell A., Mazzeo R. and Melrose R.B. (1995). Analytic surgery and the accumulation of eigenvalues. Commun. Anal. Geom. 3: 115–222 MATHMathSciNetGoogle Scholar
  10. 10.
    Hassell A. and Vasy A. (2000). Symbolic functional calculus and N-body resolvent estimates. J. Funct. Anal. 173: 257–283 MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Jensen A. and Kato T. (1979). Spectral properties of Schrödinger operators and time-decay of the wave functions. Duke Math. J. 46: 583–611 MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kirsch W. and Simon B. (1988). Corrections to the classical behaviour of the number of bound states of Schrödinger operators. Ann. Phys. 183: 122–130 MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Li H.-Q. (1999). La transformée de Riesz sur les variétés coniques. J. Funct. Anal. 168(1): 145–238 MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Melrose R.B. (1993). The Atiyah–Patodi–Singer index theorem. AK Peters, Wellesley MATHGoogle Scholar
  15. 15.
    Melrose, R.B.: Differential analysis on manifolds with corners, http://www.math.mit.edu/~rbm/book.html (in preparation)
  16. 16.
    Melrose, R.B.: Pseudodifferential Operators, Corners and Singular Limits. In: Proceedings of the international congress of mathematicians, vol. I, II (Kyoto, 1990), pp. 217–234, Mathematical Society of Japan, Tokyo (1991)Google Scholar
  17. 17.
    Melrose, R.B., Sa Barreto, A.: Zero energy limit for scattering manifolds (unpublished note)Google Scholar
  18. 18.
    Melrose R.B. (1994). Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces. In: Ikawa, M. (eds) Spectral and Scattering Theory, pp. Marcel Dekker, New York Google Scholar
  19. 19.
    Melrose R.B. (1992). Calculus of conormal distributions on manifolds with corners. Int. Math. Res. Not. 3: 51–61 CrossRefMathSciNetGoogle Scholar
  20. 20.
    Murata, M.: Asymptotic expansions in time for solutions of Schrödinger-type equations. 49, 10–56 (1982)Google Scholar
  21. 21.
    Ouhabaz E.M. (2005). Analysis of heat equations on domains. London Mathematical Society Monographs Series 31. Princeton University Press, New Jersey Google Scholar
  22. 22.
    Shen, Z.: L p estimates for Schrödinger operators with certain potentials. Ann. Inst. Fourier 45(no. 2), 513–546Google Scholar
  23. 23.
    Stein E. (1993). Harmonic Analysis. Princeton University Press, Princeton MATHGoogle Scholar
  24. 24.
    Wang, X.-P.: Asymptotic expansion in time of the Schrödinger group on conical manifolds. Ann. Inst. Fourier (Grenoble) 56(no. 6), 1903–1945 (2006), Ann. Inst. Fourier (2006) (to appear)Google Scholar

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© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Laboratoire J.-A. DieudonnéU.M.R. 6621 du C.N.R.S., Université de NiceNice Cedex 02France
  2. 2.Department of MathematicsAustralian National UniversityCanberraAustralia

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