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

[math]\displaystyle{ (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]\displaystyle{ 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]\displaystyle{ \xi \in \mathbb{R}^d }[/math]

[math]\displaystyle{ \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]\displaystyle{ s \gt 0 }[/math], the Bessel potential on [math]\displaystyle{ \mathbb{R}^d }[/math] can be represented by

[math]\displaystyle{ (I - \Delta)^{-s/2} u = G_s \ast u, }[/math]

where the Bessel kernel [math]\displaystyle{ G_s }[/math] is defined for [math]\displaystyle{ x \in \mathbb{R}^d \setminus \{0\} }[/math] by the integral formula

[math]\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. }[/math]

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

[math]\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. }[/math]

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

[math]\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}. }[/math]

Asymptotics

At the origin, one has as [math]\displaystyle{ \vert x\vert \to 0 }[/math],

[math]\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 \lt s \lt d, }[/math]

[math]\displaystyle{ G_d (x) = \frac{1}{2^{d - 1} \pi^{d/2} }\ln \frac{1}{\vert x \vert}(1 + o (1)) , }[/math]

[math]\displaystyle{ G_s (x) = \frac{\Gamma (\frac{s - d}{2})}{2^s \pi^{s/2} }(1 + o (1)) \quad \text{ if }s \gt d. }[/math]

In particular, when [math]\displaystyle{ 0 \lt s \lt d }[/math] the Bessel potential behaves asymptotically as the Riesz potential.

At infinity, one has, as [math]\displaystyle{ \vert x\vert \to \infty }[/math],

[math]\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)). }[/math]

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