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

${\displaystyle (I-\Delta )^{-s/2))$

where Δ is the Laplace operator and the fractional power is defined using Fourier transforms.

Yukawa potentials are particular cases of Bessel potentials for ${\displaystyle s=2}$ in the 3-dimensional space.

Representation in Fourier space

The Bessel potential acts by multiplication on the Fourier transforms: for each ${\displaystyle \xi \in \mathbb {R} ^{d))$

${\displaystyle {\mathcal {F))((I-\Delta )^{-s/2}u)(\xi )={\frac ((\mathcal {F))u(\xi )}{(1+4\pi ^{2}\vert \xi \vert ^{2})^{s/2))}.}$

Integral representations

When ${\displaystyle s>0}$, the Bessel potential on ${\displaystyle \mathbb {R} ^{d))$ can be represented by

${\displaystyle (I-\Delta )^{-s/2}u=G_{s}\ast u,}$

where the Bessel kernel ${\displaystyle G_{s))$ is defined for ${\displaystyle x\in \mathbb {R} ^{d}\setminus \{0\))$ by the integral formula ^[1]

${\displaystyle G_{s}(x)={\frac {1}{(4\pi )^{s/2}\Gamma (s/2)))\int _{0}^{\infty }{\frac {e^{-{\frac {\pi \vert x\vert ^{2)){y))-{\frac {y}{4\pi )))){y^{1+{\frac {d-s}{2))))}\,\mathrm {d} y.}$

Here ${\displaystyle \Gamma }$ denotes the Gamma function. The Bessel kernel can also be represented for ${\displaystyle x\in \mathbb {R} ^{d}\setminus \{0\))$ by^[2]

${\displaystyle G_{s}(x)={\frac {e^{-\vert x\vert )){(2\pi )^{\frac {d-1}{2))2^{\frac {s}{2))\Gamma ({\frac {s}{2)))\Gamma ({\frac {d-s+1}{2)))))\int _{0}^{\infty }e^{-\vert x\vert t}{\Big (}t+{\frac {t^{2)){2)){\Big )}^{\frac {d-s-1}{2))\,\mathrm {d} t.}$

This last expression can be more succinctly written in terms of a modified Bessel function,^[3] for which the potential gets its name:

${\displaystyle G_{s}(x)={\frac {1}{2^{(s-2)/2}(2\pi )^{d/2}\Gamma ({\frac {s}{2)))))K_{(d-s)/2}(\vert x\vert )\vert x\vert ^{(s-d)/2}.}$

Asymptotics

At the origin, one has as ${\displaystyle \vert x\vert \to 0}$,^[4]

${\displaystyle G_{s}(x)={\frac {\Gamma ({\frac {d-s}{2)))}{2^{s}\pi ^{s/2}\vert x\vert ^{d-s))}(1+o(1))\quad {\text{ if ))0<s<d,}$

${\displaystyle G_{d}(x)={\frac {1}{2^{d-1}\pi ^{d/2))}\ln {\frac {1}{\vert x\vert ))(1+o(1)),}$

${\displaystyle G_{s}(x)={\frac {\Gamma ({\frac {s-d}{2)))}{2^{s}\pi ^{s/2))}(1+o(1))\quad {\text{ if ))s>d.}$

In particular, when ${\displaystyle 0<s<d}$ the Bessel potential behaves asymptotically as the Riesz potential.

At infinity, one has, as ${\displaystyle \vert x\vert \to \infty }$, ^[5]

${\displaystyle G_{s}(x)={\frac {e^{-\vert x\vert )){2^{\frac {d+s-1}{2))\pi ^{\frac {d-1}{2))\Gamma ({\frac {s}{2)))\vert x\vert ^{\frac {d+1-s}{2))))(1+o(1)).}$