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Approximation and fast calculation of non-local boundary conditions for the
 

Summary: Approximation and fast calculation of
non-local boundary conditions for the
time-dependent Schr¨odinger equation
Anton Arnold1
, Matthias Ehrhardt2
, and Ivan Sofronov3
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

  

Source: Arnold, Anton - Institut für Analysis und Scientific Computing, Technische Universität Wien

 

Collections: Mathematics