Properties Of Solutions To Dirac Type Equations In Superspace

Superspaces, which were developed during the second half of the previous century, are spaces equipped with both a set of commuting variables and a set of anti-commuting variables (generating the so-called Grassmann algebra) used to describe the properties of bosons and fermions in Quantum Mechanics respectively. Therefore these superspaces and correspond-ing supermanifolds now are used extensively in theoretical physics, in subjects such as super-stringtheory, supergravitytheory and so on. As far as we know, F. A. Berezin and others studied superspaces from the point of view of algebraic geometry; on the other hand, B. DeWitt and others studied them from the point of view of differential geometry. In recent years, F. Som-men, H. DeBie and others have studied them from the point of view of functions theory, and established the fundamental framework of Clifford analysis in superspace.Based on the work of F. Sommen et al., we mainly investigate the basic properties of solutions of Dirac equations and their related functions in superspace. Moreover, we study the Almansi type expansion for solutions of Dirac type equations in superspace and their relevant content. The dissertation consists of five parts.Chapter1. We introduce the research background and research contents of this thesis. Su-perspaces contain anti-commuting variables, which are the generalization of European space Rm. As far as we know, there are two kinds of mathematical methods to study the space in history. First of all, by means of algebraic geometry, one can study that differential manifold with graded structure by introducing graded algebras based on superspace. Another important approach is based on differential geometry, that is to construct super differential manifold in superspace. Furthermore, one can prove that these two approaches to superspace are equivalent in a categorical sense. Recently, F. Sommen and coworkers drew Clifford algebra into super-space, and built the fundamental framework of Clifford analysis in superspace, so as to develop the fundamental theory in Clifford analysis to superspace. Clifford analysis is a hypercomplex function theory of functions defined in European space Rm and taken values in Clifford algebra. Correspondingly, they investigated a function theory of functions defined in superspace Rm|2n and taken values in combined function space with scalar algebra and Clifford algebra(standard orthogonal Clifford algebra and Weyl algebra). They defined generalized differential opera-tors, such as a super Dirac operator, a super Laplace operator, and a super Euler operator, etc. They constructed the fundamental solutions for all natural powers of the super Dirac operator. They brought in Berezin integral to define integral in superspace. One of the main contents of Clifford analysis is to study the function-theoretical properties of the null-solutions of Dirac equation (monogenic functions), such as Stokes theorem, Cauchy-Pompeiu formula, as well as Morera theorem, Painleve theorem and uniqueness theorem for monogenic functions so on. Correspondingly, they proved that these fundamental theorems in superspace, such as Stokes theorem, Cauchy-Pompeiu formula, Morera theorem for monogenic functions. Based on these conclusions, we further research superspace.Chapter2. We discuss the basic properties of solutions of Dirac equations and their related functions in superspace. First of all, we introduce the definition of superspace and related concept. We show the definition of a general vector variable, a general function space, general differential operators, a general integral. Next, we obtain Painleve theorem for monogenic functions in superspace by Morera theorem in superspace so that the domain of monogenic functions is enlarged; we gain uniqueness theorem for monogenic functions in superspace from Fischer decomposition in superspace; we obtain Cauchy-Riemann type equation in superspace due to the definition of the super Dirac operator. We construct monogenic functions by harmonic functions, and achieve the relationship between monogenic functions and harmonic functions, between k-monogenic functions (the null-solutions of k order Dirac equations) and monogenic functions, between k-harmonic functions (the null-solutions of k order Laplace equations) and harmonic functions, by means of the commutation rules between the differential operators in superspace(common rules in superspace); besides, we get high order Cauchy-Pompeiu formula in superspace by Stokes theorem. In particular, when the function is a k-monogenic function, it becomes Cauchy formula for k-monogenic function in superspace.Chapter3. We consider the Almansi type expansion for k-monogenic functions in super-space and its applications inspired by the Almansi expansion in Clifford analysis obtained by H. Malonek and others. The Almansi type expansion for k-mongenic function in superspace, is a k-mongenic function developed into the finite summation of the product of the monogenic functions and the correspondent natural powers of super vector-valued functions, so that the two functions are built the relationship. This lays the foundation for transforming the prob-lems for k-monogenic functions to monogenic functions and extending the conclusions for monogenic functions to k-monogenic functions in superspace. The work we do for this is as follows:first of all, we define a integral operator, a differential operator (the generalization of the super Euler operator) in star domain in superspace; Next, we prove the the inverse relation-ship between these two operators. Furthermore, we talk about the invariance of the monogenic under these two operators. Finally, we prove that the operator composed by natural powers of the super Dirac operator, operator xk, and integral operator is identity operator for the case of monogenic functions. Based on our above work, the key step of obtaining the result is to trans-form the conclusion into the problem of the direct decomposition in k-monogenic functions space, that is to decompose k-monogenic functions space into the direct-sum of the product of a monogenic function space with the correspondent powers of a super vector-valued function. In addition, we consider the applications of the Almansi type expansion for k-mongenic func-tions in superspace:the direct decomposition of polyharmonic functions space, Fischer decom-position (the foundation of spherical harmonic function theory) for homogeneous polynomial functions in star domain in superspace, as well as Painleve theorem and uniqueness theorem for k-monogenic functions in superspace because of Painleve theorem and uniqueness theorem for monogenic functions in superspace in last charpter.Chapter4. We research the Almansi type expansion for the null-solutions of Dirac type equations by means of constructing differential operator O-normalized system in superspace. B.A. Bondarenko, V.V. Karachik, Ren Guangbin and others studied f-normalized system of functions with respect to differential operators early or late, and applied these to consider the Al-mansi expansion, partial differential equations, boundary value problem. We bring in an integral operator, and establish the relationship between this integral operator and a differential operator so as to construct O-normalized system of functions with respect to Laplace operators in super-space. We obtain the Almansi type expansion for the null-solutions of polyharmonic equations in superspace by this system, so that we build the relationship between k-harmonic functions and harmonic functions in superspace. We also establish the relationship between Dirichlet pronlem (boundary value problem for the harmonic equation) and Riquier problem (boundary value problem for the polyharmonic equation), so that Riquier problem can be transformed into Dirichlet pronlem. Besides, we obtain the form null-solution of Helmohltz equations. Since Dirac operator is a factor of Laplace operator, we obtain O-normalized system of functions with respect to Dirac operators in superspace by more complex calculating. Furthermore, we get the Almansi type expansion for the formal null-solutions of k order Dirac equations, and the form null-solution of correct Dirac equations (corresponding to Helmholtz equations) by this system. When the series in the solution converges, the solution is a classical solution.Chapter5. We get the approximation to Almansi type expansion for the null-solutions of k order Dirac equations in superspace, making use of the properties of Ti operator related to Teodorescu operator. Teodorescu operator (in short T operator), a kind of singular integral operator. In the complex plane, it is the (right) inverse operator of Cauchy-Riemann operator. T operator is an important part of classic Vekua theory. Vekua theory has been widely used in elastic mechanics, thin shell theory and air dynamics, etc. Scholars at home and abroad have studied the properties of T operators in the complex plane, high dimensional complex spaces, Quaternion analysis, Clifford analysis early or late. In particular, H. Begehr, Zhang Zhongxiang and others defined Ti operator (the generalization of T operator) in Clifford analysis, and dis-cussed its elementary properties. We define Ti operator in superspace, establish the relationship between Ti operator and Cauchy type integral, show the relationship between Ti operator and Ti-1operator, and obtain the null-solutions of inhomogeneous k order Dirac equations. The last but the most important is to obtain the approximation to Almansi type expansion for the null-solutions of k order Dirac equations. That is k-monogenic functions can be expressed by the finite sum of Ti operator acting on corresponding monongenic functions, where monongenic functions can be obtained by k-monogenic functions. Furthermore, we get the necessary and sufficient condition realizing this expansion. Similarly, we define Πi operator in superspace and obtain the approximation to Almansi type expansion for the null-solutions of k order Harmonic equations in superspace.