Approximation and fast calculation of non-local boundary conditions for the time-dependent Schr¨ odinger equation Anton Arnold 1 , Matthias Ehrhardt 2 , and Ivan Sofronov 3 1 Universit¨ at M¨ unster, Institut f¨ ur Numerische Mathematik, Einsteinstr. 62, D-48149 M¨ unster, Germany (http://www.math.uni-muenster.de/u/arnold/) 2 Technische Universit¨ at Berlin, Institut f¨ ur Mathematik, Str. des 17. Juni 136, D-10623 Berlin, Germany (http://www.math.tu-berlin.de/~ehrhardt/) 3 Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, Russia (sofronov@spp.keldysh.ru) Summary. We present a way to efficiently treat the well-known transparent bound- ary conditions for the Schr¨ odinger equation. Our approach is based on two ideas: firstly, to derive a discrete transparent boundary condition (DTBC) based on the Crank-Nicolson finite difference scheme for the governing equation. And, secondly, to approximate the discrete convolution kernel of DTBC by sum-of-exponentials for a rapid recursive calculation of the convolution. We illustrate the efficiency of the proposed method on several examples. A much more detailed version of this article can be found in Arnold et al. [2003]. 1 Introduction Discrete transparent boundary conditions for the discrete 1D–Schr¨ odinger equation −iR(ψ j,n+1 − ψ j,n )= ∆ 2 (ψ j,n+1 + ψ j,n ) − wV j,n+ 1 2 (ψ j,n+1 + ψ j,n ) , (1) where ∆ 2 ψ j = ψ j +1 − 2ψ j + ψ j −1 , R =4∆x 2 /∆t, w =2∆x 2 , V j,n+ 1 2 := V (x j ,t n+ 1 2 ), x j = j∆x, j ∈ ZZ; and V (x,t)= V − = const. for x ≤ 0; V (x,t)= V + = const. for x ≥ X, t ≥ 0, ψ(x, 0) = ψ I (x), with supp ψ I ⊂ [0,X], were introduced in Arnold [1998]. The DTBC at e.g. the left boundary point j =0 reads, cf. Thm. 3.8 in Ehrhardt and Arnold [2001]: ψ 1,n − s 0 ψ 0,n = ∑ n−1 k=1 s n−k ψ 0,k − ψ 1,n−1 ,n ≥ 1. (2) The convolution kernel {s n } can be obtained by explicitly calculating the inverse Z–transform of the function ˆ s(z) := z+1 z ˆ ℓ 0 (z), where ˆ ℓ 0 (z)=1 − iζ ±  −ζ (ζ +2i), ζ = R 2 z−1 z+1 + i∆x 2 V − (choose sign such that | ˆ ℓ 0 (z)| > 1).