The one-body density matrix of the pure system at zero temperature
is given by (1.47). One can rewrite this integral as
(6.1)
with the function defined as
(6.2)
and
being the healing length.
Let us integrate (B.1) by parts
times. All terms which have the form
disappear, because
the sinus function is equal to zero in and
, terms with cosine contribute
.
The result of the integration is the following
(6.3)
The -th derivative of the function can be calculated from
the differentiation of the Taylor expansion of in zero
(6.4)
As a result one has a representation of in series of
powers of .
By using the expansion (B.3) one can calculate the
leading terms of for
(6.5)
Which means that the asymptotic behavior of
is one over distance squared6.1.
This behavior is correct at distances where the contribution from the
second term in (B.5) can be neglected, i.e.
which means distances
much larger than the healing length.
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