In mathematics, the Bessel potential is a potential (named after Friedrich Wilhelm Bessel) similar to the Riesz potential but with better decay properties at infinity.
If s is a complex number with positive real part then the Bessel potential of order s is the operator
(I-\Delta)-s/2
Yukawa potentials are particular cases of Bessel potentials for
s=2
The Bessel potential acts by multiplication on the Fourier transforms: for each
\xi\inRd
l{F}((I-\Delta)-s/2u)(\xi)=
l{F | |
u |
(\xi)}{(1+4\pi2\vert\xi\vert2)s/2
When
s>0
Rd
(I-\Delta)-s/2u=Gs\astu,
Gs
x\inRd\setminus\{0\}
Gs(x)=
1 | |
(4\pi)s/2\Gamma(s/2) |
infty | |
\int | |
0 |
| ||||||||||||
|
dy.
\Gamma
x\inRd\setminus\{0\}
Gs(x)=
e-\vert | ||||||||||||||||||||||
|
infty | |
\int | |
0 |
e-\vert(t+
t2 | |
2 |
| ||||
) |
dt.
This last expression can be more succinctly written in terms of a modified Bessel function,[3] for which the potential gets its name:
G | |||||||||||||
|
K(d-s)/2(\vertx\vert)\vertx\vert(s-d)/2.
At the origin, one has as
\vertx\vert\to0
Gs(x)=
| ||||||
2s\pis/2\vertx\vertd |
(1+o(1)) if0<s<d,
Gd(x)=
1 | |
2d\pid/2 |
ln
1 | |
\vertx\vert |
(1+o(1)),
Gs(x)=
| ||||||
2s\pis/2 |
(1+o(1)) ifs>d.
0<s<d
At infinity, one has, as
\vertx\vert\toinfty
Gs(x)=
e-\vert | ||||||||||||||||||||||||||
|
(1+o(1)).