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the overline denotes the complex conjugate, and
z
m
D
a
m
C
ib
m
are arbitrary
complex numbers with Im
z
2
m
/
¤
0
. In its general form, the polynomial (
8.21
)is
additionally multiplied by the product of the arbitrary number of real negative roots
(
.
). The ensuing analysis of (
8.21
) will be simplified by omitting the product
(summation) limits over
b
m
D
0
and assuming there are no real negative or multiple roots.
The latter requirement is not restrictive in practice, because location of the roots is
never known exactly, and the BEC spectrum can always be well approximated by
(
8.21
)(
Yaremchuk and Sentchev 2012
).
It is instructive to note that the polynomial (
8.21
) can also be rewritten as
m
Y
B
1
.k
2
/
D
1
Z
.a
m
C
.k
b
m
/
2
/.a
m
C
.k
C
b
m
/
2
/;
(8.23)
m
D
1
Compared to the spectral representation (
8.20
), representation (
8.23
)hasthe
advantage that its free parameters are not constrained by the positive-definiteness
requirement, and they have a sensible meaning of the scales (
b
1
) and “energies”
a
1
) of the modes forming the spectrum.
Using (
8.6
), the matrix elements of
B
can now be written as
(
Z
B
n
.r/
D
Zr
s
.2/
k
s
C
1
J
s
.kr/dk
Q
m
.k
2
C
z
2
m
/.k
2
C
z
2
m
/
;
(8.24)
n
2
0
s
D
n=2
1
where
. The integral in (
8.24
) can be taken by decomposing
Z
B.k/
D
Q
m
.k
2
C
z
2
m
/.k
2
C
z
2
m
/
(8.25)
into elementary fractions:
B.k/
D
X
m
q
m
k
2
C
z
2
m
q
m
k
2
C
z
2
m
C
;
(8.26)
where
Z
q
m
D
.
z
2
m
z
2
m
/
Q
(8.27)
z
2
m
z
j
/.
z
2
m
z
j
/
j
¤
m
.
After substitution of (
8.26
)into(
8.24
), the integral is reduced to the sum of
Hankel-Nicholson type integrals (
Abramowitz and Stegun 1972
) and can be taken
explicitly, yielding
X
B
n
.r/
D
2r
2
n
.2/
h
q
m
s
m
K
s
.
m
/
i
(8.28)
n
2
m
m
D
z
m
r
where
, and angular brackets denote taking the real part (cf. (
8.13
)).
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