Research Article

Semigroup Forum

, Volume 41, Issue 1, pp 97-114

First online:

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

  • M. M. H. PangAffiliated withDepartment of Mathematics, King’s College London

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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}.