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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.

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

$ℱ\left(\right(I-\Delta {)}^{-s/2}u\left)\right(\xi )=\frac{ℱu\left(\xi \right)}{(1+4{\pi }^2|\xi {|}^2{)}^{s/2}}.$

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

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

where the Bessel kernel $G_s$ is defined for $x\in {ℝ}^d\setminus \left\{0\right\}$ by the integral formula ^[1]

$G_s\left(x\right)=\frac{1}{\left(4\pi {)}^{s/2}\Gamma \right(s/2)}{\int }_0^{\infty }\frac{e^{-\frac{\pi |x{|}^2}{y}-\frac{y}{4\pi }}}{y^{1+\frac{d-s}{2}}}dy.$

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

$G_s\left(x\right)=\frac{e^{-\left|x\right|}}{\left(2\pi {)}^{\frac{d-1}{2}}{2}^{\frac{s}{2}}\Gamma \right(\frac{s}{2}\left)\Gamma \right(\frac{d-s+1}{2})}{\int }_0^{\infty }{e}^{-\left|x\right|t}(t+\frac{t^2}{2}{)}^{\frac{d-s-1}{2}}dt.$

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

$G_s\left(x\right)=\frac{1}{2^{(s-2)/2}\left(2\pi {)}^{d/2}\Gamma \right(\frac{s}{2})}{K}_{(d-s)/2}\left(\right|x\left|\right)|x{|}^{(s-d)/2}.$

At the origin, one has as $\left|x\right|\to 0$,^[4]

$G_s\left(x\right)=\frac{\Gamma \left(\frac{d-s}{2}\right)}{2^s{\pi }^{s/2}|x{|}^{d-s}}(1+o(1\left)\right)\text{ if }0<s<d,$

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

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

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

At infinity, one has, as $\left|x\right|\to \infty$,^[5]

$G_s\left(x\right)=\frac{e^{-\left|x\right|}}{2^{\frac{d+s-1}{2}}{\pi }^{\frac{d-1}{2}}\Gamma \left(\frac{s}{2}\right)|x{|}^{\frac{d+1-s}{2}}}(1+o(1\left)\right).$

• Riesz potential
• Fractional integration
• Sobolev space
• Fractional Schrödinger equation
• Yukawa potential

• Duduchava, R. (2001) [1994], "Bessel potential operator", Encyclopedia of Mathematics, EMS Press
• Grafakos, Loukas (2009), Modern Fourier analysis, Graduate Texts in Mathematics, vol. 250 (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-0-387-09434-2, ISBN 978-0-387-09433-5, MR 2463316, S2CID 117771953
• Hedberg, L.I. (2001) [1994], "Bessel potential space", Encyclopedia of Mathematics, EMS Press
• Solomentsev, E.D. (2001) [1994], "Bessel potential", Encyclopedia of Mathematics, EMS Press
• Stein, Elias (1970), Singular integrals and differentiability properties of functions, Princeton, NJ: Princeton University Press, ISBN 0-691-08079-8