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
:$(I-\Delta)^$
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 $\xi \in \mathbb^d$
:$\mathcal((I-\Delta)^ u) (\xi)= \frac.$

Integral representations

When $s > 0$, the Bessel potential on $\mathbb^d$ can be represented by
:$(I - \Delta)^ u = G_s \ast u,$
where the Bessel kernel $G_s$ is defined for $x \in \mathbb^d \setminus \$ by the integral formula
:$G_s (x) = \frac \int_0^\infty \frac\,\mathrmy.$
Here $\Gamma$ denotes the Gamma function.
The Bessel kernel can also be represented for $x \in \mathbb^d \setminus \$ by
:$G_s (x) = \frac \int_0^\infty e^ \Big(t + \frac\Big)^\frac \,\mathrmt.$

This last expression can be more succinctly written in terms of a modified Bessel function, for which the potential gets its name:
:$G_s(x)=\fracK_(\vert x \vert) \vert x \vert^.$

Asymptotics

At the origin, one has as $\vert x\vert \to 0$,
:$G_s (x) = \frac(1 + o (1)) \quad \text 0 < s < d,$
:$G_d (x) = \frac\ln \frac(1 + o (1)) ,$
:$G_s (x) = \frac(1 + o (1)) \quad \texts > d.$
In particular, when $0 < s < d$ the Bessel potential behaves asymptotically as the Riesz potential.

At infinity, one has, as $\vert x\vert \to \infty$,
:$G_s (x) = \frac(1 + o (1)).$

See also

* Riesz potential
* Fractional integration
* Sobolev space
* Fractional Schrödinger equation
* Yukawa potential

References

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* {{citation , first=Elias , last=Stein , authorlink=Elias Stein , title=Singular integrals and differentiability properties of functions , publisher=Princeton University Press , location=Princeton, NJ , year=1970 , isbn=0-691-08079-8 , url-access=registration , url=https://archive.org/details/singularintegral0000stei

Fractional calculus
Partial differential equations
Potential theory
Singular integrals