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.

If s is a complex number with positive real part then the Bessel potential of order s is the operator

( I − Δ ) − s / 2 {\displaystyle (I-\Delta )^{-s/2}} {\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 s = 2 {\displaystyle s=2} {\displaystyle 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 {\displaystyle \xi \in \mathbb {R} ^{d}} {\displaystyle \xi \in \mathbb {R} ^{d}}

F ( ( I − Δ ) − s / 2 u ) ( ξ ) = F u ( ξ ) ( 1 + 4 π 2 | ξ | 2 ) s / 2 . {\displaystyle {\mathcal {F}}((I-\Delta )^{-s/2}u)(\xi )={\frac {{\mathcal {F}}u(\xi )}{(1+4\pi ^{2}\vert \xi \vert ^{2})^{s/2}}}.} {\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 s > 0 {\displaystyle s>0} {\displaystyle s>0}, the Bessel potential on R d {\displaystyle \mathbb {R} ^{d}} {\displaystyle \mathbb {R} ^{d}} can be represented by

( I − Δ ) − s / 2 u = G s ∗ u , {\displaystyle (I-\Delta )^{-s/2}u=G_{s}\ast u,} {\displaystyle (I-\Delta )^{-s/2}u=G_{s}\ast u,}

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

G s ( x ) = 1 ( 4 π ) s / 2 Γ ( s / 2 ) ∫ 0 ∞ e − π | x | 2 y − y 4 π y 1 + d − s 2 d y . {\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.} {\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 Γ {\displaystyle \Gamma } {\displaystyle \Gamma } denotes the Gamma function. The Bessel kernel can also be represented for x ∈ R d ∖ { 0 } {\displaystyle x\in \mathbb {R} ^{d}\setminus \{0\}} {\displaystyle x\in \mathbb {R} ^{d}\setminus \{0\}} by[2]

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 . {\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.} {\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:

G s ( x ) = 1 2 ( s − 2 ) / 2 ( 2 π ) d / 2 Γ ( s 2 ) K ( d − s ) / 2 ( | x | ) | x | ( s − d ) / 2 . {\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}.} {\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 | x | → 0 {\displaystyle \vert x\vert \to 0} {\displaystyle \vert x\vert \to 0},[4]

G s ( x ) = Γ ( d − s 2 ) 2 s π s / 2 | x | d − s ( 1 + o ( 1 ) )  if  0 < s < d , {\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_{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 ) = 1 2 d − 1 π d / 2 ln ⁡ 1 | x | ( 1 + o ( 1 ) ) , {\displaystyle G_{d}(x)={\frac {1}{2^{d-1}\pi ^{d/2}}}\ln {\frac {1}{\vert x\vert }}(1+o(1)),} {\displaystyle G_{d}(x)={\frac {1}{2^{d-1}\pi ^{d/2}}}\ln {\frac {1}{\vert x\vert }}(1+o(1)),}
G s ( x ) = Γ ( s − d 2 ) 2 s π s / 2 ( 1 + o ( 1 ) )  if  s > d . {\displaystyle G_{s}(x)={\frac {\Gamma ({\frac {s-d}{2}})}{2^{s}\pi ^{s/2}}}(1+o(1))\quad {\text{ if }}s>d.} {\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 0 < s < d {\displaystyle 0<s<d} {\displaystyle 0<s<d} the Bessel potential behaves asymptotically as the Riesz potential.

At infinity, one has, as | x | → ∞ {\displaystyle \vert x\vert \to \infty } {\displaystyle \vert x\vert \to \infty },[5]

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

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.