Solutions of the Poisson Equation and Related Equations in Super Spinor Space
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Solutions of the Poisson Equation and Related Equations in Super Spinor Space Hongfen Yuan1
Received: 31 March 2015 / Revised: 16 March 2016 / Accepted: 3 April 2016 / Published online: 28 May 2016 © Springer-Verlag Berlin Heidelberg 2016
Abstract In this paper, we construct solutions to the polyharmonic equation, the Helmholtz equation, the Poisson equation and the inhomogeneous polyharmonic equation by the 0-normalized system of functions with respect to the Laplace operator in super spinor space. In an analogous way, applying the 0-normalized system of functions with respect to the Dirac operator in super spinor space, we obtain solutions to the polyDirac equation, the modified Dirac equation and the inhomogeneous Dirac equation. Keywords Super spinor space · Normalized systems of functions · Helmholtz equation · Poisson equation · Inhomogeneous polyharmonic equation Mathematics Subject Classification Primary 58C50; Secondary 31A30
1 Introduction In 2013, Coulembier [1] constructed the super spinors Sm|2n (i.e., the spinors for the orthosymplectic superalgebra osp(m|2n)), which generalizes the spinors for so(m) [2] and the symplectic spinors for sp(2n) [3]. The super spinors play an important part
Communicated by Stephan Ruscheweyh. Research supported by the TianYuan Special Funds of the National Natural Science Foundation of China [No. 11426082], the Natural Science Foundation of Hebei Province [No. A2016402034] and Project of Handan Municipal Science and Technology Bureau [No. 1534201097-10].
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Hongfen Yuan [email protected] College of Science, Hebei University of Engineering, Handan 056038, China
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in the development of Lie superalgebra, where Lie superalgebras are used in geometry, and also in classical and quantum mechanics. Moreover, Coulembier and De Bie [4] defined the Dirac operator ∂x , acting on super functions defined on R m|2n with values in super spinor space Sm|2n , which generalizes the Cauchy–Riemann operators by Stein and Weiss [5]. It is interesting that the Dirac operator is the natural extension of both the classical Dirac operator, for the case n = 0, which acts on the functions defined R m with values in the orthogonal spinors Sm [6], and the symplectic Dirac operator, for the case m = 0, which acts on sp(2n) on differential forms on R 2n with values in the symplectic spinors S0|2n [7]. Then, in [8] the authors defined the Laplace operator ∂x2 and studied the commuting relations between differential operators in super spinor space. Based on this work, we consider some differential equations in super spinor space by normalized systems of functions. Normalized systems of functions were advocated by Bondarenko [9]. Karachik [10] constructed 0-normalized system of functions with respect to a Laplace operator in R m . Applying the system, he constructs solutions of the Helmholtz equation, the Poisson equation and so on (see [11–13]). In this paper, we mainly investigate solutions of the polyharmonic equation, the Helmholtz equation, the Poisson equation and the inhomogene
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