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

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Abstract

Let −Δ be the Dirichlet Laplacian onR N and letV be a potential satisfyingV +∈K Nloc 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}.

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  1. Department of Mathematics, King’s College London, Strand, WC2R 2LS, London, England

    M. M. H. Pang

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  1. M. M. H. Pang
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Pang, M.M.H. Resolvent estimates for schrödinger operators inL P(RN) and the theory of exponentially boundedC-semigroups. Semigroup Forum 41, 97–114 (1990). https://doi.org/10.1007/BF02573381

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