Applications of complex Clifford analysis to the study of solutions to generalized Dirac and Klein-Gordon equations with holomorphic potentials
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 \({\mathbb{C}}^ n\).
scientific article; zbMATH DE number 4051992
scientific article