Advances in Applied Clifford Algebras

, Volume 25, Issue 3, pp 755–769 | Cite as

Polynomial Dirac Operators in Superspace

Article

Abstract

In this paper, we construct fundamental solutions for the polynomial operators \({{(\partial_{x}-\lambda)^l}, \prod \limits_{k=1}^{l} (\partial_{x}- \lambda_{k})^{n_k},\, \partial^{l}_{x}+ \sum \limits_{j=1}^{l} b_j \partial^{l-j}_{x}}\), where \({{\partial_{x}}}\) is the Dirac operator in superspace, and \({{\lambda, \lambda_{k}, b_j}}\) are complex numbers. Applying the fundamental solutions, we investigate Cauchy-Pompeiu formulas for the operators \({{{(\partial_{x}- \lambda)^l}, \partial^{l}_{x} + \sum \limits_{j=1}^{l} b_j \partial^{l-j}_{x}}}\). In addition, we obtain a Cauchy-Pompeiu formula for the operator \({\prod \limits_{k=1}^{l} (\partial_{x}- \lambda_{k})^{n_k}}\) and a Cauchy formula for the operator \({\prod \limits_{k=1}^{l} (\partial_{x}- \lambda_{k})}\) by another way.

Keywords

Superspace polynomial Dirac operator fundamental solution Cauchy-Pompeiu formula 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Dirac P. A. M.: The Quantum theory of the Electron. Proc. R. Soc. Lond. A. 117, 610–624 (1928)MATHCrossRefADSGoogle Scholar
  2. 2.
    Brack F., Delanghe R., Sommen F: Clifford analysis. Res Notes Math, Pitman London (1982)Google Scholar
  3. 3.
    H. De Bie, F. Sommen, Correct rules for Clifford calculus on superspace. Adv. appl. Clifford alg.17 (2007), 357–382.Google Scholar
  4. 4.
    H. De Bie, F. Sommen, Fundamental solutions for the super Laplace and Dirac operators and all their natural powers. J. Math Anal.Appl.338 (2008), 1320– 1328.Google Scholar
  5. 5.
    H. De Bie, F. Sommen, Spherical harmonics and integration in superspace. J. Phys. A: Math. Theor. 40 (26) (2007), 7193–7212.Google Scholar
  6. 6.
    K. Coulembier, H. De Bie, F. Sommen, Integration in superspace using distribution theory. J. Phys. A: Math. Theor. 42 (2009), 395206 (23pp).Google Scholar
  7. 7.
    F. Sommen, Z. Y. Xu, Fundamental solutions for operators which are polynomials in the Dirac operator. Kluwer. Acad. Publ. Dordrecht, 1992.Google Scholar
  8. 8.
    J. Ryan, Cauchy-Green type formulae in Clifford analysis. Transactions of the American Mathematical society 347(4) (1995), 1331–1341.Google Scholar
  9. 9.
    Y. F. Gong, T. Qian, J. Y. Du, Structure of Solutions of Polynomial Dirac Equations in Clifford Analysis. Complex Variables 49(1) (2003), 15–24.Google Scholar
  10. 10.
    F. A. Berezin, Introduction to algebra and analysis with anticommuting variables. Moskov. Gos. Univ. Moscow, 1983.Google Scholar
  11. 11.
    N. Aronszajn, T. M. Creese, L. J. Lipkin, Polyharmonic functions. Oxford Mathematics Monographs, The Clarendon Press, Oxford University Press, New York, 1983.Google Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.College of ScienceHebei University of EngineeringHandanPR China
  2. 2.College of Mathemstics and Information ScienceHebei Normal UniversityShijiazhuangPR China

Personalised recommendations