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Bessel potential

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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.

Contents

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 ∈ R d

F ( ( I − Δ ) − s / 2 u ) ( ξ ) = F u ( ξ ) ( 1 + 4 π 2 | ξ | 2 ) s / 2 .

Integral representations

When s > 0 , the Bessel potential on R d can be represented by

( I − Δ ) − s / 2 = G s ∗ u ,

where the Bessel kernel G s is defined for x ∈ R d ∖ { 0 } by the integral formula

G s ( x ) u = 1 ( 4 π ) s / 2 Γ ( s / 2 ) ∫ 0 ∞ e − π | x | 2 δ − δ 4 π δ 1 + d − s 2 d δ .

Here Γ denotes the Gamma function. The Bessel kernel can also be represented for x ∈ R d ∖ { 0 } by

G s ( x ) = e − | x | ( 2 π ) d − 1 2 2 s 2 Γ ( s 2 ) Γ ( d − s + 1 2 ) ∫ 0 ∞ e − | x | t ( t + t 2 2 ) d − s − 1 2 d t .

Asymptotics

At the origin, one has as | x | → 0 ,

G s ( x ) = Γ ( d − s 2 ) 2 s π s / 2 | x | n − s ( 1 + o ( 1 ) )  if  0 < s < d , G s ( x ) = 1 2 d − 1 π d / 2 ln ⁡ 1 | x | ( 1 + o ( 1 ) ) , G s ( x ) = Γ ( 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 | → ∞ ,

G s ( x ) = e − | x | 2 d + s − 1 2 π d − 1 2 Γ ( s 2 ) | x | n + 1 − s 2 ( 1 + o ( 1 ) ) .

References

Bessel potential Wikipedia
(Text) CC BY-SA


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