Pseudodifferential operator

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.

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 first proof of the Atiyah–Singer index theorem. 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 ${\displaystyle u}$ 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:

${\displaystyle \quad P(D)u(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}\int _{\mathbb {R} ^{n}}e^{i(x-y)\xi }P(\xi )u(y)\,dy\,d\xi }$

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