Discretization of Continuous Spectra Based on Perfectly Matched Layers

Abstract

As a tool of analysis in physics, wavefields are often expanded in a set of eigensolutions obtained from a Sturm--Liouville problem. For singular Sturm--Liouville problems subject to radiation boundary conditions, i.e., problems defined on an infinite domain, this set of eigensolutions has continuous parts. In this paper we will show that it is possible to approximate this continuous set of eigensolutions by a discrete set of eigensolutions of the same Sturm--Liouville operator but subject to Dirichlet boundary conditions in complex space. The idea of Dirichlet boundary conditions in complex space stems from the perfectly matched layer (PML) absorbing boundary condition. The PML was introduced in 1994 [J. P. Bérenger, J. Comput. Phys., 114 (1994), pp. 185--200] as an absorbing termination of a finite difference time domain grid. These complex space Dirichlet boundary conditions have been used recently to close open electromagnetic waveguide structures. In the present paper we aim at developing a mathematical basis for the wavefields existing in such structures. On the one hand, this yields a better understanding of the properties of such waveguides and their applications in electromagnetic field problems. On the other hand, this opens the road for applications in other wavefields such as elastodynamics and quantum mechanics.