{{Short description|Mathematical 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 <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. 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,<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.
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 * Fractional integration * Sobolev space * Fractional Schrödinger equation *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 | location=Berlin, New York | edition=2nd | series=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 |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