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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 \(\|(z-\Delta+V)^{-1}\|_{p,p}\), \(\hbox {Im} z\neq 0\).

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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[1] Arendt, W.,Vector valued Laplace transforms and Cauchy problems, Israel J. Math.,59, (1987), 327–352. · Zbl 0637.44001 · doi:10.1007/BF02774144
[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). · Zbl 0675.45017
[3] Cycon, H. L., Froese, R. G., Kirsch, W. and Simon, B.,Schrödinger operators, Texts and Monographs in Physics, Springer-Verlag, 1987. · Zbl 0619.47005
[4] Davies, E. B.,Kernel estimates for functions of second order elliptic operators, Quart. J. Math. Oxford (2)39, (1988), 37–46. · Zbl 0652.35024 · doi:10.1093/qmath/39.1.37
[5] Davies, E. B.,Pointwise bounds on the space and time derivatives of heat kernels, to appear. · Zbl 0702.35106
[6] Davies, E. B.,Heat Kernels and Spectral Theory Camb. Univ. Press, 1989. · Zbl 0699.35006
[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. · Zbl 0651.47026 · doi:10.1112/plms/s3-55.1.181
[8] Hormander, L.,Estimates for translation invariant operators in L p spaces. Acta Math.104 (1960), 93–140. · Zbl 0093.11402 · doi:10.1007/BF02547187
[9] Kellerman, H.,Integrated semigroups, J. Funct. Anal., to appear. · Zbl 0689.47014
[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. · Zbl 0167.10401
[11] Miyadera, I.,On the generators of exponentially bounded C-semigroups, Proc. Japan Acad. Series A, Vol.62 (1986), 239–242. · Zbl 0617.47032 · doi:10.3792/pjaa.62.239
[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. · Zbl 0642.47034
[13] Neubrander, F.,Integrated semigroups and their applications to the abstract Cauchy problem, Pac. J. Math.135 (1988), 111–155. · Zbl 0675.47030
[14] Simon, B.,Schrödinger semigroups, Bull. Amer. Math. Soc.7 (1982), 447–526. · Zbl 0524.35002 · doi:10.1090/S0273-0979-1982-15041-8
[15] Sjöstrand, S.,On the Riesz means of the solutions of the Schrödinger equation, Ann. Scuola Norm. Sup. Pisa24 (1970), 331–348. · Zbl 0201.14901
[16] Tanaka, N.,On the exponentially bounded C-semigroups, Tokyo J. Math.10 (1987), 107–117. · Zbl 0631.47029 · doi:10.3836/tjm/1270141795
[17] Davies, E. B.,Some norm bounds and quadratic form inequalities for Schrödinger operators, J. Operator Theory,9 (1983), 147–162. · Zbl 0516.47025
[18] Davies, E. B.,Heat kernel bounds for second order elliptic operators on Riemannian manifolds, Amer. J. Math.109 (1987), 545–570. · Zbl 0648.58037 · doi:10.2307/2374567
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