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

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

where Δ is the Laplace operator and the fractional power is defined using Fourier transforms.

Yukawa potentials are particular cases of Bessel potentials for ${\displaystyle s=2}$ in the 3-dimensional space.

Representation in Fourier space

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

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

Integral representations

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

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

where the Bessel kernel ${\displaystyle G_{s}}$ is defined for ${\displaystyle x\in \mathbb {R} ^{d}\setminus \{0\}}$ by the integral formula ^[1]

${\displaystyle G_{s}(x)={\frac {1}{(4\pi )^{s/2}\Gamma (s/2)}}\int _{0}^{\infty }{\frac {e^{-{\frac {\pi \vert x\vert ^{2}}{y}}-{\frac {y}{4\pi }}}}{y^{1+{\frac {d-s}{2}}}}}\,\mathrm {d} y.}$

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

${\displaystyle G_{s}(x)={\frac {e^{-\vert x\vert }}{(2\pi )^{\frac {d-1}{2}}2^{\frac {s}{2}}\Gamma ({\frac {s}{2}})\Gamma ({\frac {d-s+1}{2}})}}\int _{0}^{\infty }e^{-\vert x\vert t}{\Big (}t+{\frac {t^{2}}{2}}{\Big )}^{\frac {d-s-1}{2}}\,\mathrm {d} t.}$

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

${\displaystyle G_{s}(x)={\frac {1}{2^{(s-2)/2}(2\pi )^{d/2}\Gamma ({\frac {s}{2}})}}K_{(d-s)/2}(\vert x\vert )\vert x\vert ^{(s-d)/2}.}$

Asymptotics

At the origin, one has as ${\displaystyle \vert x\vert \to 0}$,^[4]

${\displaystyle G_{s}(x)={\frac {\Gamma ({\frac {d-s}{2}})}{2^{s}\pi ^{s/2}\vert x\vert ^{d-s}}}(1+o(1))\quad {\text{ if }}0<s<d,}$

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

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

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

At infinity, one has, as ${\displaystyle \vert x\vert \to \infty }$, ^[5]

${\displaystyle G_{s}(x)={\frac {e^{-\vert x\vert }}{2^{\frac {d+s-1}{2}}\pi ^{\frac {d-1}{2}}\Gamma ({\frac {s}{2}})\vert x\vert ^{\frac {d+1-s}{2}}}}(1+o(1)).}$

See also

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

References

• 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