The discretization of the Wigner potential operator has already been implicitly discussed in the previous section on conservation of mass. We have shown, that one has to use an equispaced product grid to obtain conservation of mass in the discrete case.
Another undesirable feature of the potential operator is that
we need the potential outside the simulation domain
in the calculation of
from 8.9,
assuming that the integration domain is symmetric around
.
Usually the potential is extended outside the domain with
the values at the boundary which are fixed by the applied
bias. With this we get that the integrand
in the calculation of
is equal to the applied bias for
and,
in general
. Consequently
has a
singularity at
= 0. As
is odd,
it is natural to set
.
The singularity can be calculated analytically, which reduces to
the Fourier transform of a Heaviside function.
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