# Bessel potential

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{{Short description|Mathematical potential}}
In [mathematics](/source/mathematics), the '''Bessel potential''' is a [potential](/source/potential_theory) (named after [Friedrich Wilhelm Bessel](/source/Friedrich_Wilhelm_Bessel)) similar to the [Riesz potential](/source/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](/source/Laplace_operator) and the [fractional power](/source/fractional_calculus) is defined using Fourier transforms.

[Yukawa potential](/source/Yukawa_potential)s 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 transform](/source/Fourier_transform)s: 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 <ref>{{cite book|last1=Stein|first1=Elias|title=Singular integrals and differentiability properties of functions|url=https://archive.org/details/singularintegral0000stei|url-access=registration|date=1970|publisher=Princeton University Press|isbn=0-691-08079-8|at=Chapter V eq. (26)}}</ref>
:<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](/source/Gamma_function).
The Bessel kernel can also be represented for <math>x \in \mathbb{R}^d \setminus \{0\} </math> by<ref>{{cite journal|last1=N. Aronszajn|last2=K. T. Smith|title=Theory of Bessel potentials I|journal=Ann. Inst. Fourier|date=1961|volume=11|doi=10.5802/aif.116 |at=385–475, (4,2)|doi-access=free}}</ref>
:<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](/source/Bessel_function),<ref>{{cite journal|last1=N. Aronszajn|last2=K. T. Smith|title=Theory of Bessel potentials I|journal=Ann. Inst. Fourier|date=1961|volume=11|doi=10.5802/aif.116 |at=385–475|doi-access=free}}</ref> 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>,<ref>{{cite journal|last1=N. Aronszajn|last2=K. T. Smith|title=Theory of Bessel potentials I|journal=Ann. Inst. Fourier|date=1961|volume=11|doi=10.5802/aif.116 |at=385–475, (4,3)|doi-access=free}}</ref>
:<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](/source/Riesz_potential).

At infinity, one has, as <math>\vert x\vert \to \infty </math>,<ref>{{cite journal|last1=N. Aronszajn|last2=K. T. Smith|title=Theory of Bessel potentials I|journal=Ann. Inst. Fourier|date=1961|volume=11|pages=385–475|doi=10.5802/aif.116 |doi-access=free}}</ref>
:<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](/source/Riesz_potential)
* [Fractional integration](/source/Fractional_integration)
* [Sobolev space](/source/Sobolev_space)
* [Fractional Schrödinger equation](/source/Fractional_Schr%C3%B6dinger_equation)
*[Yukawa potential](/source/Yukawa_potential)

==References==
{{Reflist}}
*{{eom|id=B/b110420|title=Bessel potential operator|first=R. |last=Duduchava}}
*{{Citation | last1=Grafakos | first1=Loukas | title=Modern Fourier analysis | publisher=[Springer-Verlag](/source/Springer-Verlag) | location=Berlin, New York | edition=2nd | series=[Graduate Texts in Mathematics](/source/Graduate_Texts_in_Mathematics) | isbn=978-0-387-09433-5 | doi=10.1007/978-0-387-09434-2 | mr=2463316 | year=2009 | volume=250| s2cid=117771953 }}
*{{eom|title=Bessel potential space|first=L.I. |last= Hedberg}}
*{{eom|title=Bessel potential|first=E.D.|last= Solomentsev}}
* {{citation |first=Elias |last=Stein |authorlink=Elias Stein |title=Singular integrals and differentiability properties of functions |publisher=[Princeton University Press](/source/Princeton_University_Press) |location=Princeton, NJ |year=1970 |isbn=0-691-08079-8 |url-access=registration |url=https://archive.org/details/singularintegral0000stei }}

Category:Fractional calculus
Category:Partial differential equations
Category:Potential theory
Category:Singular integrals

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Adapted from the Wikipedia article [Bessel potential](https://en.wikipedia.org/wiki/Bessel_potential) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Bessel_potential?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.