Bessel potential

Template:Short description 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 - Δ)^{- 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 $ξ \in ℝ^{d}$

ℱ ((I - Δ)^{- s / 2} u) (ξ) = \frac{ℱ u (ξ)}{(1 + 4 π^{2} | ξ |^{2})^{s / 2}} .

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

(I - Δ)^{- s / 2} u = G_{s} * u,

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

G_{s} (x) = \frac{1}{(4 π)^{s / 2} Γ (s / 2)} \int_{0}^{\infty} \frac{e^{- \frac{π | x |^{2}}{y} - \frac{y}{4 π}}}{y^{1 + \frac{d - s}{2}}} d y .

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

G_{s} (x) = \frac{e^{- | x |}}{(2 π)^{\frac{d - 1}{2}} 2^{\frac{s}{2}} Γ (\frac{s}{2}) Γ (\frac{d - s + 1}{2})} \int_{0}^{\infty} e^{- | x | t} (t + \frac{t^{2}}{2})^{\frac{d - s - 1}{2}} 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 π)^{d / 2} Γ (\frac{s}{2})} K_{(d - s) / 2} (| x |) | x |^{(s - d) / 2} .

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

G_{s} (x) = \frac{Γ (\frac{d - s}{2})}{2^{s} π^{s / 2} | x |^{d - s}} (1 + o (1)) if 0 < s < d,

G_{d} (x) = \frac{1}{2^{d - 1} π^{d / 2}} \ln \frac{1}{| x |} (1 + o (1)),

G_{s} (x) = \frac{Γ (\frac{s - d}{2})}{2^{s} π^{s / 2}} (1 + o (1)) if s > d .

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

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

G_{s} (x) = \frac{e^{- | x |}}{2^{\frac{d + s - 1}{2}} π^{\frac{d - 1}{2}} Γ (\frac{s}{2}) | x |^{\frac{d + 1 - s}{2}}} (1 + o (1)) .