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Complex Analysis and Operator Theory

, Volume 8, Issue 2, pp 461–484 | Cite as

The Schrödinger Semigroup on Some Flat and Non Flat Manifolds

  • R. S. Kraußhar
  • M. M. Rodrigues
  • N. VieiraEmail author
Article

Abstract

In this paper we apply known techniques from semigroup theory to the Schrödinger problem with initial conditions. To this end, we define the regularized Schrödinger semigroup acting on a space-time domain and show that it is strongly continuous and contractive in \(L_p,\) with \(\frac{3}{2}\!<\!p\!<\!3.\) These results can easily be extended to the case of conformal operators acting in the context of differential forms, but they require positiveness conditions on the curvature of the considered Minkowski manifold. For that purpose, we will use a Clifford algebra setting in order to highlight the geometric characteristics of the manifold. We give an application of such methods to the regularized Schrödinger problem with initial condition and we will extended our conclusions to the limit case. For the torus case and a class of non-oriented higher dimensional Möbius strip like domains we also give some explicit formulas for the fundamental solution.

Keywords

Clifford analysis Semigroup theory Schrödinger equation  Dissipative operators Hypoelliptic operators 

Mathematics Subject Classification (2000)

Primary 30G35 Secondary 47H06 35H10 

Notes

Acknowledgments

The second and third authors were supported by FEDER founds through COMPETE-operational programme factors of competitiveness (“programa operacional factores de competitividade”) and by Portuguese funds through the Center for Research and Development in Mathematics and Applications (University of Aveiro) and the Portuguese Foundation for Science and Technology (“FCT—Fundação para a Ciência e a Tecnologia”), within project PEst-C/MAT/UI4106 /2011 with COMPETE number FCOMP-01-0124-FEDER-022690.

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Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Fachbereich Mathematik, Arbeitsgruppe AlgebraTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Department of Mathematics, CIDMA-Center for Research and Development in Mathematics and ApplicationsUniversity of AveiroAveiroPortugal

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