Stable Perfectly Matched Layers for the Schrödinger Equations
We present a well-posed and stable perfectly matched layer (PML) for the time-dependent Schrödinger wave equations. The layer consists of the Hamiltonian (H) perturbed by a complex absorbing potential (CAP, − iσ1), H cap = H − iσ1, and carefully chosen auxiliary functions to ensure a zero reflection coefficient at the interface between the physical domain and the layer. Using standard perturbation techniques, we show that the layer is asymptotically stable. Numerical experiments are presented showing the accuracy of the new PML model. The numerical scheme couples the standard Crank–Nicolson scheme for the modified wave equation to an explicit scheme of the Runge–Kutta type for the auxiliary differential equations.
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