Multi-periodic eigensolutions to the Dirac operator and applications to the generalized Helmholtz equation on flat cylinders and on the n-torus

摘要

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 multi-periodic eigensolutions to the Dirac operator associated with a complex parameter lambda is an element of 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.