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Concept
Pseudo-differential operator
Summary
In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory, e.g. in mathematical models that include ultrametric pseudo-differential equations in a non-Archimedean space.
The study of pseudo-differential operators began in the mid 1960s with the work of Kohn, Nirenberg, Hörmander, Unterberger and Bokobza.
They played an influential role in the second proof of the Atiyah–Singer index theorem via K-theory. Atiyah and Singer thanked Hörmander for assistance with understanding the theory of pseudo-differential operators.
Consider a linear differential operator with constant coefficients,
which acts on smooth functions with compact support in Rn.
This operator can be written as a composition of a Fourier transform, a simple multiplication by the
polynomial function (called the symbol)
and an inverse Fourier transform, in the form:
Here, is a multi-index, are complex numbers, and
is an iterated partial derivative, where ∂j means differentiation with respect to the j-th variable. We introduce the constants to facilitate the calculation of Fourier transforms.
Derivation of formula ()
The Fourier transform of a smooth function u, compactly supported in Rn, is
and Fourier's inversion formula gives
By applying P(D) to this representation of u and using
one obtains formula ().
To solve the partial differential equation
we (formally) apply the Fourier transform on both sides and obtain the algebraic equation
If the symbol P(ξ) is never zero when ξ ∈ Rn, then it is possible to divide by P(ξ):
By Fourier's inversion formula, a solution is
Here it is assumed that:
P(D) is a linear differential operator with constant coefficients,
its symbol P(ξ) is never zero,
both u and ƒ have a well defined Fourier transform.
The last assumption can be weakened by using the theory of distributions.
The first two assumptions can be weakened as follows.
Official source
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