**
Discrete transparent boundary conditions for the 1D Schrödinger equation**
**scaled Schrödinger equation:**
with
for
for
**goal:** Solve the whole space problem (almost) exactly on the
*computational interval* by introducing *transparent boundary conditions* at
.
**Crank-Nicolson finite difference scheme:**
grid points:
(with
),
approximation:
;
numerical scheme for whole space problem:
**discrete transparent boundary conditions:**
(to be used with scheme (1) on
)
with convolution kernels
:
... Legendre polynomials (
)
initial condition must satisfy:
REMARK: The evaluation of the convolutions (2) is very expensive for large-time
calculations
approximative "sum-of-exponential'' convolution
coefficients strongly reduce the numerical effort.
**approximative transparent boundary conditions:**
(3) |
... parameter to choose
Java-applet for calculation of ,
for given
download Maple-code for calculation of
**fast evaluation of discrete convolutions:**
For the "sum-of-exponential'' convolution coefficients (3) the resulting
convolution in (2):
can be computed very efficiently by the algorithm:
where
**example - free Schrödinger equation:**
Gaussian beam, travelling right
Solution with L=10:
Solution with
L=20:
[Arnold - Ehrhardt - Sofronov '02] |