Bessel potential

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

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

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

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

\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 $$s > 0$$, the Bessel potential on $$\mathbb{R}^d$$ can be represented by
 * $$(I - \Delta)^{-s/2} u = G_s \ast u,$$

where the Bessel kernel $$G_s$$ is defined for $$x \in \mathbb{R}^d \setminus \{0\} $$ by the integral formula

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

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, for which the potential gets its name:

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 $$\vert x\vert \to 0 $$,

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, $$

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

G_s (x) = \frac{\Gamma (\frac{s - d}{2})}{2^s \pi^{s/2} }(1 + o (1)) \quad \text{ if }s > d. $$ In particular, when $$0 < s < d$$ the Bessel potential behaves asymptotically as the Riesz potential.

At infinity, one has, as $$\vert x\vert \to \infty $$,

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)). $$