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.
[1] 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.
[2] Consider a linear differential operator with constant coefficients, which acts on smooth functions
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,
are complex numbers, and is an iterated partial derivative, where ∂j means differentiation with respect to the j-th variable.
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: The last assumption can be weakened by using the theory of distributions.
For instance, if P(x,ξ) is an infinitely differentiable function on Rn × Rn with the property for all x,ξ ∈Rn, all multiindices α,β, some constants Cα, β and some real number m, then P belongs to the symbol class
This means that one can solve linear elliptic differential equations more or less explicitly by using the theory of pseudo-differential operators.
The singularity of the kernel on the diagonal depends on the degree of the corresponding operator.