Perfectly matched layers for Schrödinger-type equations with nontrivial energy–momentum dispersion

Abstract

The construction of perfectly-matched-layer (PML) boundary conditions is extended to wave equations with non-trivial energy–momentum dispersion relation. Complex coordinate extension is based upon the relative sign between the surface-normal component of group velocity and the k-vector. This ensures selective damping of propagating and counter-propagating modes at the simulation boundaries. Moreover, the technical complexity of the PML method can be inferred from the energy–momentum dispersion relation of the underlying wave equation, respectively, the finite-difference scheme employed. The PML transform is derived for selected Schrödinger-type (i.e. first-order-in-time) kinetic transport equations. Spectral analysis demonstrates proper damping behaviour of out-propagating modes. Results are presented for the Schrödinger equation for particles and holes, the s- and p-wave Bogoliubov–de Gennes equation, the Dirac equation, and the Dohnal k⋅p model. Where available, a comparison to previous PML formulations is made.