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.

Representation in Fourier space

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

\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 s > 0, the Bessel potential on \mathbb{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 \mathbb{R}^d \setminus \{0\} by the integral formula [1]

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

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:

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 \vert x\vert \to 0,[4]

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,
G_d (x) = \frac{1}{2^{d - 1} \pi^{d/2} }\ln \frac{1}{\vert x \vert}(1 + o (1)) ,
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 0 < s < d the Bessel potential behaves asymptotically as the Riesz potential.

At infinity, one has, as \vert x\vert \to \infty,[5]

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

References

  1. ^ Stein, Elias (1970). Singular integrals and differentiability properties of functions. Princeton University Press. Chapter V eq. (26). ISBN 0-691-08079-8.
  2. ^ N. Aronszajn & K. T. Smith (1961). "Theory of Bessel potentials I". Ann. Inst. Fourier. 11: 385–475, (4,2). doi:10.5802/aif.116
  3. ^ N. Aronszajn & K. T. Smith (1961). "Theory of Bessel potentials I". Ann. Inst. Fourier. 11: 385–475. doi:10.5802/aif.116
  4. ^ N. Aronszajn & K. T. Smith (1961). "Theory of Bessel potentials I". Ann. Inst. Fourier. 11: 385–475, (4,3). doi:10.5802/aif.116
  5. ^ N. Aronszajn & K. T. Smith (1961). "Theory of Bessel potentials I". Ann. Inst. Fourier. 11: 385–475. doi:10.5802/aif.116