Semigroup Forum

, Volume 41, Issue 1, pp 97–114 | Cite as

Resolvent estimates for schrödinger operators inL P (R N ) and the theory of exponentially boundedC-semigroups

  • M. M. H. Pang
Research Article


Let −Δ be the Dirichlet Laplacian onR N and letV be a potential satisfyingV +K loc N andV ∈K N . Using the Gaussian upper bound for the heat kernel ofe (Δ−v)t we obtain estimates for growth of ‖(z−Δ+V)−1 p,p in the region {z: Im(z)≠0} and show that Δ−V generates an (N+2)-times integrated semigroup onL p (R N ), 1≤p≤∞. A sharper estimate for the resolvent is obtained ifV is further assumed to be either in\(\{ V:\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{V} \in L^1 (R^N )\} \) or {V:V(x)≥c|x| 4N/(N+2)+c for all |x|≥P>0}.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Arendt, W.,Vector valued Laplace transforms and Cauchy problems, Israel J. Math.,59, (1987), 327–352.MATHMathSciNetGoogle Scholar
  2. [2]
    Arendt, W. and Kellermann, H.,Integrated solutions of Volterra integrodifferential equations and applications. In: Integro-differential Equations. Proc. Conf. Trento 1987. G. Da Prato, M. Iannelli (eds.), Pitman (to appear).Google Scholar
  3. [3]
    Cycon, H. L., Froese, R. G., Kirsch, W. and Simon, B.,Schrödinger operators, Texts and Monographs in Physics, Springer-Verlag, 1987.Google Scholar
  4. [4]
    Davies, E. B.,Kernel estimates for functions of second order elliptic operators, Quart. J. Math. Oxford (2)39, (1988), 37–46.MATHCrossRefGoogle Scholar
  5. [5]
    Davies, E. B.,Pointwise bounds on the space and time derivatives of heat kernels, to appear.Google Scholar
  6. [6]
    Davies, E. B.,Heat Kernels and Spectral Theory Camb. Univ. Press, 1989.Google Scholar
  7. [7]
    Davies, E. B., Pang, M.M.H.,The Cauchy problem and a generalization of the Hille-Yosida theorem, Proc. London Math. Soc.55, (1987), 181–208.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Hormander, L.,Estimates for translation invariant operators in L p spaces. Acta Math.104 (1960), 93–140.MathSciNetGoogle Scholar
  9. [9]
    Kellerman, H.,Integrated semigroups, J. Funct. Anal., to appear.Google Scholar
  10. [10]
    Lanconelli, E.,Valutazioni in L p R N)della solutione del problema di Cauchy per l’equazione di Schrödinger, Boll. Un. Mat. Ital.4 (1968), 591–607.MathSciNetGoogle Scholar
  11. [11]
    Miyadera, I.,On the generators of exponentially bounded C-semigroups, Proc. Japan Acad. Series A, Vol.62 (1986), 239–242.MATHMathSciNetGoogle Scholar
  12. [12]
    Miyadera, I. and Tanaka, N.,Some remarks on the exponentially bounded C-semigroups and the integrated semigroups, Proc. Japan Acad.63, Series A (1987), 139–142.MATHMathSciNetGoogle Scholar
  13. [13]
    Neubrander, F.,Integrated semigroups and their applications to the abstract Cauchy problem, Pac. J. Math.135 (1988), 111–155.MATHMathSciNetGoogle Scholar
  14. [14]
    Simon, B.,Schrödinger semigroups, Bull. Amer. Math. Soc.7 (1982), 447–526.MATHMathSciNetCrossRefGoogle Scholar
  15. [15]
    Sjöstrand, S.,On the Riesz means of the solutions of the Schrödinger equation, Ann. Scuola Norm. Sup. Pisa24 (1970), 331–348.MATHMathSciNetGoogle Scholar
  16. [16]
    Tanaka, N.,On the exponentially bounded C-semigroups, Tokyo J. Math.10 (1987), 107–117.MATHMathSciNetCrossRefGoogle Scholar
  17. [17]
    Davies, E. B.,Some norm bounds and quadratic form inequalities for Schrödinger operators, J. Operator Theory,9 (1983), 147–162.MATHMathSciNetGoogle Scholar
  18. [18]
    Davies, E. B.,Heat kernel bounds for second order elliptic operators on Riemannian manifolds, Amer. J. Math.109 (1987), 545–570.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag New York Inc 1990

Authors and Affiliations

  • M. M. H. Pang
    • 1
  1. 1.Department of MathematicsKing’s College LondonLondonEngland

Personalised recommendations