Applications of complex Clifford analysis to the study of solutions to generalized Dirac and Klein-Gordon equations with holomorphic potentials
type: Item
instance of: scholarly article
title: Applications of complex Clifford analysis to the study of solutions to generalized Dirac and Klein-Gordon equations with holomorphic potentials
zbMATH Open document ID: 0645.32002
DOI: 10.1016/0022-0396(87)90129-X
author: B. George
published in: Journal of Differential Equations
publication date: 1987-01-01T00:00:00Z
review text: Suppose that, for \(\Omega \subseteq {\mathbb{C}}^ n\), \(B: \Omega \to A_ n({\mathbb{C}})\) is a holomorphic function, and \(A_ n({\mathbb{C}})\) is the complex, \(2^ n\) dimensional Clifford algebra. The first part of the paper deals with constructing holomorphic solutions to the equation \(DA=B\), where D is the homogeneous Dirac operator in \({\mathbb{C}}^ n\). The solutions are constructed by homotopically deforming singular integrals, over real, n-dimensional manifolds, within \({\mathbb{C}}^ n\). These techniques are then applied to locally solve the differential equation \(DA=CA\) where C is a holomorphic function. The techniques use local isomorphisms and enables the holomorphic continuations of the function A to be determined. Similar techniques are applied to the equation \(\Delta A=CA\), where \(\Delta\) is the Laplacian in \({\mathbb{C}}^ n\).
Mathematics Subject Classification ID: 32A30
Mathematics Subject Classification ID: 30G35
Mathematics Subject Classification ID: 15A66
Mathematics Subject Classification ID: 32D15
Mathematics Subject Classification ID: 44A10
Mathematics Subject Classification ID: 44A15
zbMATH DE Number: 4051992
zbMATH Keywords: complex Clifford algebra
zbMATH Keywords: holomorphic continuation
zbMATH Keywords: complex Laplacian
zbMATH Keywords: homogeneous Dirac operator
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4051992