Advances in Applied Clifford Algebras

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

Polynomial Dirac Operators in Superspace



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.


Superspace polynomial Dirac operator fundamental solution Cauchy-Pompeiu formula 


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© 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

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