Absorbing boundary conditions for time-dependent Schrödinger equations: A density-matrix formulation

This paper presents some absorbing boundary conditions for simulations based on the time-dependent Schrödinger equations. The boundary conditions are expressed in terms of the elements of the density-matrix, and it is derived from the full model over a much larger domain. To make the implementation much more efficient, several approximations for the convolution integral will be constructed with guaranteed stability. These approximations lead to modified density-matrix equations at the boundary. The effectiveness is examined via numerical tests.