Complex Analysis and Operator Theory

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

The Schrödinger Semigroup on Some Flat and Non Flat Manifolds

Article
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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 
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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.

References

  1. 1.
    Artino, R., Barros-Neto, J.: Hypoelliptic boundary-value problems, lectures notes in pure and applied mathematics, vol. 53. Marcel Dekker, New York, Bassel (1980)Google Scholar
  2. 2.
    Berezin, F.A., Shubin, M.A.: The Schrödinger equation, mathematics and its applications, vol. 66. Kluwer Academic Publishers, Dordrecht (1991)CrossRefGoogle Scholar
  3. 3.
    Cerejeiras, P., Vieira, N.: Regularization of the non-stationary Schrödinger operator. Math. Meth. Appl. Sci. 32(4), 535–555 (2009)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Charlier, A., Bérard, A., Charlier, M.F., Fristot, D.: Tensors and the Clifford algebra, pure and applied mathematics, vol. 163. Marcel Dekker, New York (1992)Google Scholar
  5. 5.
    Delanghe, R., Sommen, F., Souček, V.: Clifford algebras and spinor-valued functions, mathematics and its applications, vol. 53. Kluwer Academic Publishers, Dordrecht (1992)CrossRefGoogle Scholar
  6. 6.
    Duduchava, L.R., Mitrea, D., Mitrea, M.: Differential operators and boundary value problems on hypersurfaces. Math. Nachr. 279(9–10), 996–1023 (2006)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Eichhorn, J.: The heat semigroup acting on tensors or differential forms with values in vector bundle. Arch. Math. 27(1), 15–24 (1991)MathSciNetGoogle Scholar
  8. 8.
    Evans, L.C.: Partial differential equations, graduate studies in mathematics, vol. 19. American Mathematical Society, Providence (1997)Google Scholar
  9. 9.
    Friedrich, T.: Zur Abhangigheit des Dirac-operators von der spin-Struktur. Colloq. Math. 48, 57–62 (1984)MATHMathSciNetGoogle Scholar
  10. 10.
    Gilbert, J.E., Murray, M.: Clifford algebras and Dirac operators in harmonic analysis, Cambridge studies in advanced mathematics, vol. 26. Cambridge University Press, Cambridge (1991)CrossRefGoogle Scholar
  11. 11.
    Graaf, J.: Evolution equations, textos de matemtica-Série B-No. 14, Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade de Coimbra (1998)Google Scholar
  12. 12.
    Grafarend, E.W.: Tensor algebra, linear algebra, multilinear algebras. Technical reports, Department of Geodesy and Geoinformatics, University of Stuttgart, Stuttgart (2004)Google Scholar
  13. 13.
    Günter, N.: Potential theory and its application to the basic problems of mathematical physics. Frederick Ungar Publishing, New York (1967)Google Scholar
  14. 14.
    Habala, P., Hájek, P., Zizler, V.: Introduction to Banach spaces I. Matfyzpress, Vydavatelství Matematicko-fyzikální fakulty Univerzity Karlovy (1996)Google Scholar
  15. 15.
    Isakov, V.: Inverse problems for partial differential equations, applied mathematical sciences, vol. 127. Springer, New York (2006)Google Scholar
  16. 16.
    Kraußhar, R.S., Ryan, J.: Some conformally flat spin manifolds, Dirac operators and automorphic forms. J. Math. Anal. Appl. 325(1), 359–376 (2007)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Kraußhar, R.S., Vieira, N.: The Schrödinger equation on cylinders and the \(n\)-torus. J. Evol. Equ. 11(1), 215–237 (2011)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Kuiper, N.H.: On conformally flat spaces in the large. Ann. Math. 2(50), 916–924 (1949)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Leutwiler, H.: Remarks on modified Clifford analysis, potential theory—ICPT’94. In: Král, J. (ed.) Proceedings of the international conference, Kouty, Czech Republic, 13–20 Aug, deGruyter, Berlin, pp. 389–397 (1996)Google Scholar
  20. 20.
    Miatello, R., Podesta, R.: Spin structures and spectra of \(Z_{2}\) manifolds. Math. Z. 247, 319–335 (2004)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Pfäffle, F.: The Dirac spectrum of Bieberbach manifolds. J. Geom. Phys. 35, 367–385 (2000)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Reed, M., Simon, B.: Methods of modern mathematical physics-vol. II, fourier analysis, self-adjointness. Academic Press, New York (1975)Google Scholar
  23. 23.
    Ryan, J.: Cauchy kernels for some conformally flat manifolds. In: Trends in mathematics: advances in analysis and geometry, pp. 149–160. Birkhäuser, Basel (2004)Google Scholar
  24. 24.
    Schrödinger, E.: An undulatory theory of the mechanics of atoms and molecules. Phy. Rev. 28(6), 1049–1070 (1926)CrossRefMATHGoogle Scholar
  25. 25.
    Tao, T.: Nonlinear dispersive equations, local and global analysis, CBMS regional conference series in mathematics, vol. 106. American Mathematical Society, Providence (2006)Google Scholar
  26. 26.
    Velo, G.: Mathematical aspects of the nonlinear Schrödinger equation. In: Liis, Vázquez, et al. (eds.) Proceedings of the Euroconference on nonlinear Klein-Gordon and Schrödinger systems: theory and applications, pp. 39–67. World Scientific, Singapore (1996)Google Scholar
  27. 27.
    Chung, L.K., Zhao, Z.: From Brownian motion to Schrödinger equation, Grundlehren der Mathematischen Wissenschaften, vol. 312. Springer, Berlin (1995)CrossRefGoogle Scholar

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© 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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