Rolf Sören Kraußhar

Fock spaces, Landau operators and the time-harmonic Maxwell equations

We investigate the representations of the solutions to Maxwells equations based on the combination of hypercomplex function-theoretical methods with quantum mechanical methods. Our approach provides us with a characterization for the solutions to the time-harmonic Maxwell system in terms of series expansions involving spherical harmonics resp. spherical monogenics. Also, a thorough investigation for the series representation of the solutions in terms of eigenfunctions of Landau operators that encode n-dimensional spinless electrons is given. This new insight should lead to important investigations in the study of regularity and hypo-ellipticity of the solutions to Schrodinger equations with natural applications in relativistic quantum mechanics concerning massive spinor fields.

On the Klein-Gordon Equation on Some Examples of Conformally Flat Spin 3-Manifolds

In this paper we present an overview about our recent results on the analytic treatment of the Klein-Gordon equation on some conformally flat 3-tori and on 3-spheres.

Basics on growth orders of polymonogenic functions

In this paper, we introduce generalizations of the classical growth order and the growth type of analytic functions in the context of polymonogenic functions. Polymonogenic functions are null-solutions of higher integer order iterates of a generalized higher dimensional Cauchy–Riemann operator. One of the main goals is to prove generalizations of the famous Lindelöf–Pringsheim theorem linking explicitly these growth orders and growth types with the Taylor series coefficients in the context of this function class.

A new class of hypercomplex analytic cusp forms

In this paper we deal with a new class of Clifford algebra valued automorphic forms on arithmetic subgroups of the Ahlfors-Vahlen group. The forms that we consider are in the kernel of the operator D Delta(k/2) for some even k is an element of Z. They will be called k-holomorphic Cliffordian automorphic forms. k-holomorphic Cliffordian functions are well equipped with many function theoretical tools. Furthermore, the real component functions also have the property that they are solutions to the homogeneous and inhomogeneous Weinstein equations. This function class includes the set of k-hypermonogenic functions as a special subset. While we have not been able so far to propose a construction for non-vanishing k-hypermonogenic cusp forms for k not equal 0, we are able to do so within this larger set of functions. After having explained their general relation to hyperbolic harmonic automorphic forms, we turn to the construction of Poincare series. These provide us with non-trivial examples of cusp forms within this function class. Then we establish a decomposition theorem of the spaces of k-holomorphic Cliffordian automorphic forms in terms of a direct orthogonal sum of the spaces of k-hypermonogenic Eisenstein series and of k-holomorphic Cliffordian cusp forms.

On the Incompressible Viscous Stationary MHD Equations and Explicit Solution Formulas for Some Three-dimensional Radially Symmetric Domai

The quaternionic calculus is a powerful tool to treat many complicated systems of linear and non-linear PDEs in higher dimensions. In this paper we apply these new techniques to treat the stationary incompressible viscous magnetohydrodynamic equations. For the highly viscous case, in which the convective terms are negligibly small we present explicit analytic representation formulas for some three-dimensional radially symmetric domains. Then we look at the fully non-linear case for which we propose a fixed point algorithm. In this more complicated context, the solutions of the simpler linear problems treated in the first part of the paper need to be used to solving the corresponding equations in each step of the proposed iteration.

Some Applications of Parabolic Dirac Operators to the Instationary Navier-Stokes Problem on Conformally Flat Cylinders and Tori in \(\mathbb {R}^3\)

In this paper we give a survey on how to apply recent techniques of Clifford analysis over conformally flat manifolds to deal with instationary flow problems on cylinders and tori. Solutions are represented in terms of integral operators involving explicit expressions for the Cauchy kernel that are associated to the parabolic Dirac operators acting on spinor sections of these manifolds.