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

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

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 $\xi \in {\mathbb{ℝ}}^d$

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

Integral representations

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

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

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

$G_s(x)=\frac{1}{(4\pi {)}^{s/2}\mathrm{\Gamma }(s/2)}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-\frac{\pi |x{|}^2}{y}-\frac{y}{4\pi }}}{y^{1+\frac{d-s}{2}}}\mathrm{d}y.$

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

$G_s(x)=\frac{e^{-|x|}}{(2\pi {)}^{\frac{d-1}{2}}{2}^{\frac{s}{2}}\mathrm{\Gamma }(\frac{s}{2})\mathrm{\Gamma }(\frac{d-s+1}{2})}\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-|x|t}(t+\frac{t^2}{2}{)}^{\frac{d-s-1}{2}}\mathrm{d}t.$

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

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

Asymptotics

At the origin, one has as $|x|\to 0$,

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

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

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

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

At infinity, one has, as $|x|\to \mathrm{\infty }$,

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

See also

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

References

• Hazewinkel, Michiel, ed. (2001), "Bessel potential operator", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=B/b110420 
• Grafakos, Loukas (2009), Modern Fourier analysis, Graduate Texts in Mathematics, 250 (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-0-387-09434-2, ISBN 978-0-387-09433-5 
• Hazewinkel, Michiel, ed. (2001), "Bessel potential space", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page 
• Hazewinkel, Michiel, ed. (2001), "Bessel potential", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page 
• Stein, Elias (1970), Singular integrals and differentiability properties of functions, Princeton, NJ: Princeton University Press, ISBN 0-691-08079-8, https://archive.org/details/singularintegral0000stei