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Resolvent estimates for Schrödinger operators in $$L^ p(R^ N)$$ and the theory of exponentially bounded $$C$$-semigroups. (English) Zbl 0739.47017
The Schrödinger equation $${\partial\psi \over\partial t}=i\Delta\psi+V\psi$$ can be studied in the spaces $$L^ p(\mathbb{R}^ N)$$, $$1\leq p\leq\infty$$. Difficulties arise if $$p\neq 2$$ because then the “ unperturbed evolution operator” $$e^{i\Delta t}$$ is unbounded. The difficulties can be overcome using a technique of Sjöstrand. Starting from these results, the author obtains growth estimates on the norm of the resolvents $$\|(z-\Delta+V)^{-1}\|_{p,p}$$, $$\hbox {Im} z\neq 0$$.

MSC:
 47D06 One-parameter semigroups and linear evolution equations 47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX) 35J10 Schrödinger operator, Schrödinger equation 47A10 Spectrum, resolvent
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References:
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