Boundary value problems associated to a Hermitian Helmholtz equation

AbstractAs is the case for the Laplace operator, in Euclidean Clifford analysis also the Helmholtz operator can be factorized, more precisely by using perturbed Dirac operators. In this paper we consider the Helmholtz equation in a circulant matrix form in the context of Hermitian Clifford analysis. The aim is to introduce and study the corresponding inhomogeneous Hermitian Dirac operators, which will constitute a splitting of the traditional perturbed Dirac operators of the Euclidean Clifford analysis context. This will not only lead to special solutions of the Hermitian Helmholtz equation as such, but also to the study of boundary value problems of Riemann type for those solutions, which are, in fact, solutions of the Hermitian perturbed Dirac operators involved

Hölder norm estimate for a Hilbert transform in Hermitian Clifford analysis

A Hilbert transform for Holder continuous circulant (2 x 2) matrix functions, on the d-summable (or fractal) boundary I" of a Jordan domain Omega in a"e(2n) , has recently been introduced within the framework of Hermitean Clifford analysis. The main goal of the present paper is to estimate the Holder norm of this Hermitean Hilbert transform. The expression for the upper bound of this norm is given in terms of the Holder exponents, the diameter of I" and a specific d-sum (d > d) of the Whitney decomposition of Omega. The result is shown to include the case of a more standard Hilbert transform for domains with left Ahlfors-David regular boundary

A Hilbert transform for hermitean matrix functions on fractal domains

We consider Holder continuous circulant (2 × 2) matrix functions defined on the fractal boundary of a Jordan domain in R2n. The main goal is to establish a Hilbert transform for such functions, within the framework of Hermitean Clifford analysis. This is a higher dimensional function theory centered around the simultaneous null solutions of two first order vector valued differential operators, called Hermitean Dirac operators. In a previous paper a Hermitean Cauchy integral was constructed by means of a matrix approach using circulant (2×2) matrix functions, from which a Hilbert transform was derived, all of this for the case of domains with smooth boundary. However, crucial parts of the method used are not extendable to the case where the boundary of the considered domain is fractal. At present we propose an alternative approach which will enable us to define a new Hermitean Hilbert transform in that case. As a consequence, we are able to give necessary and sufficient conditions for the Hermitean monogenicity of a circulant matrix function in the interior and exterior of the domain considered, in terms of its boundary value, where the boundary is required to be Ahlfors-David regular

Monogenic Functions and Representations of Nilpotent Lie Groups in Quantum Mechanics

We describe several different representations of nilpotent step two Lie groups in spaces of monogenic Clifford valued functions. We are inspired by the classic representation of the Heisenberg group in the Segal-Bargmann space of holomorphic functions. Connections with quantum mechanics are described. Keywords: Segal-Bargmann space, Heisenberg group, coherent states, wavelet transform, reproducing kernel, nilpotent Lie group, monogenic functions, Dirac operator, Clifford algebra, (second) quantization, quantum field theory.Comment: 27 pages; LaTeX2e; no picture