Bessel potential
Encyclopedia
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
where Δ is the Laplace operator
and the fractional power
is defined using Fourier transforms.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Bessel potential is a potential
Potential theory
In mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials which...
(named after Friedrich Wilhelm Bessel) similar to the Riesz potential
Riesz potential
In mathematics, the Riesz potential is a potential named after its discoverer, the Hungarian mathematician Marcel Riesz. In a sense, the Riesz potential defines an inverse for a power of the Laplace operator on Euclidean space...
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
where Δ is the Laplace operator
Laplace operator
In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or Δ...
and the fractional power
Fractional calculus
Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers or complex number powers of the differentiation operator.and the integration operator J...
is defined using Fourier transforms.
See also
- Riesz potentialRiesz potentialIn mathematics, the Riesz potential is a potential named after its discoverer, the Hungarian mathematician Marcel Riesz. In a sense, the Riesz potential defines an inverse for a power of the Laplace operator on Euclidean space...
- Fractional integration
- Sobolev spaceSobolev spaceIn mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space...
- Fractional Schrödinger equationFractional Schrödinger equationThe fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics. It was discovered by Nick Laskin as a result of extending the Feynman path integral, from the Brownian-like to Lévy-like quantum mechanical paths. The term fractional Schrödinger equation was coined by...