Title: Multiperiodic eigensolutions to the Dirac operator and applications to the generalized Helmholtz equation on flat cylinders and on the n-torus Authors: Constales, DenisKrausshar, Rolf Sören # × Issue Date: Nov-2009 Publisher: John Wiley and Sons Ltd. Series Title: Mathematical Methods in the Applied Sciences vol:32 issue:16 pages:2050-2070 Abstract: In this paper we study the solutions to the generalized Helmholtz equation with complex parameter on some conformally flat cylinders and on the $n$-torus. Using the Clifford algebra calculus, the solutions can be expressed as multiperiodic eigensolutions to the Dirac operator associated to a complex parameter $\lambda \in {\bf C}$. Physically, these can be interpreted as the solutions to the time-harmonic Maxwell equations on these manifolds. We study their fundamental properties and give an explicit representation theorem of all these solutions and develop some integral representation formulas. In particular we set up Green type formulas for the cylindrical and toroidal Helmholtz operator. As a concrete application we explicitly solve the Dirichlet problem for the cylindrical Helmholtz operator on the half-cylinder. Finally, we introduce hypercomplex integral operators on these manifolds which allow us to represent the solutions to the inhomogeneous Helmholtz equation with given boundary data on cylinders and on the $n$-torus. Description: IMPACTFACTOR 2007: 0.594 ISSN: 0170-4214 Publication status: published KU Leuven publication type: IT Appears in Collections: Analysis Section