Английская Википедия: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
- <math>(I-\Delta)^{-s/2}</math>
where Δ is the Laplace operator and the fractional power is defined using Fourier transforms.
Yukawa potentials are particular cases of Bessel potentials for <math>s=2</math> in the 3-dimensional space.
Representation in Fourier space
The Bessel potential acts by multiplication on the Fourier transforms: for each <math>\xi \in \mathbb{R}^d</math>
- <math>
\mathcal{F}((I-\Delta)^{-s/2} u) (\xi)= \frac{\mathcal{F}u (\xi)}{(1 + 4 \pi^2 \vert \xi \vert^2)^{s/2}}.
</math>
Integral representations
When <math>s > 0</math>, the Bessel potential on <math>\mathbb{R}^d</math> can be represented by
- <math>(I - \Delta)^{-s/2} u = G_s \ast u,</math>
where the Bessel kernel <math>G_s</math> is defined for <math>x \in \mathbb{R}^d \setminus \{0\} </math> by the integral formula [1]
- <math>
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.
</math> Here <math>\Gamma</math> denotes the Gamma function. The Bessel kernel can also be represented for <math>x \in \mathbb{R}^d \setminus \{0\} </math> by[2]
- <math>
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. </math>
This last expression can be more succinctly written in terms of a modified Bessel function,[3] for which the potential gets its name:
- <math>
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}. </math>
Asymptotics
At the origin, one has as <math>\vert x\vert \to 0 </math>,[4]
- <math>
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, </math>
- <math>
G_d (x) = \frac{1}{2^{d - 1} \pi^{d/2} }\ln \frac{1}{\vert x \vert}(1 + o (1)) , </math>
- <math>
G_s (x) = \frac{\Gamma (\frac{s - d}{2})}{2^s \pi^{s/2} }(1 + o (1)) \quad \text{ if }s > d. </math> In particular, when <math>0 < s < d</math> the Bessel potential behaves asymptotically as the Riesz potential.
At infinity, one has, as <math>\vert x\vert \to \infty </math>, [5]
- <math>
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)). </math>
See also
- Riesz potential
- Fractional integration
- Sobolev space
- Fractional Schrödinger equation
- Yukawa potential
References
