Pseudo-differential operator

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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.

In simpler terms, the definition of a pseudo-differential operator depends on the Fourier Transform. This topic is actually covered long after the introduction of the Fourier Transform. The Fourier Transform is covered at varying levels of rigor beginning with a cursory introduction in a standard second year differential equations course focusing on applications and progressing to a rigorous introduction often found in advanced undergraduate to beginning graduate courses in Fourier Analysis which relies on the use of mathematical analysis rather than exclusively univariate or multivariate calculus.

History[edit]

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 first proof of the Atiyah–Singer index theorem. Atiyah and Singer thanked Hörmander for assistance with understanding the theory of Pseudo-differential operators.[2]

Motivation[edit]

Linear differential operators with constant coefficients[edit]

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:

 

 

 

 

(1)

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 (1)

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 (1).

Representation of solutions to partial differential equations[edit]

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:

  1. P(D) is a linear differential operator with constant coefficients,
  2. its symbol P(ξ) is never zero,
  3. 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.

In the last formula, write out the Fourier transform of ƒ to obtain

This is similar to formula (1), except that 1/P(ξ) is not a polynomial function, but a function of a more general kind.

Definition of pseudo-differential operators[edit]

Here we view pseudo-differential operators as a generalization of differential operators. We extend formula (1) as follows. A pseudo-differential operator P(x,D) on Rn is an operator whose value on the function u(x) is the function of x:

 

 

 

 

(2)

where is the Fourier transform of u and the symbol P(x,ξ) in the integrand belongs to a certain symbol class. 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 of Hörmander. The corresponding operator P(x,D) is called a pseudo-differential operator of order m and belongs to the class

Properties[edit]

Linear differential operators of order m with smooth bounded coefficients are pseudo-differential operators of order m. The composition PQ of two pseudo-differential operators PQ is again a pseudo-differential operator and the symbol of PQ can be calculated by using the symbols of P and Q. The adjoint and transpose of a pseudo-differential operator is a pseudo-differential operator.

If a differential operator of order m is (uniformly) elliptic (of order m) and invertible, then its inverse is a pseudo-differential operator of order −m, and its symbol can be calculated. This means that one can solve linear elliptic differential equations more or less explicitly by using the theory of pseudo-differential operators.

Differential operators are local in the sense that one only needs the value of a function in a neighbourhood of a point to determine the effect of the operator. Pseudo-differential operators are pseudo-local, which means informally that when applied to a distribution they do not create a singularity at points where the distribution was already smooth.

Just as a differential operator can be expressed in terms of D = −id/dx in the form

for a polynomial p in D (which is called the symbol), a pseudo-differential operator has a symbol in a more general class of functions. Often one can reduce a problem in analysis of pseudo-differential operators to a sequence of algebraic problems involving their symbols, and this is the essence of microlocal analysis.

Kernel of pseudo-differential operator[edit]

Pseudo-differential operators can be represented by kernels. The singularity of the kernel on the diagonal depends on the degree of the corresponding operator. In fact, if the symbol satisfies the above differential inequalities with m ≤ 0, it can be shown that the kernel is a singular integral kernel.

See also[edit]

Footnotes[edit]

  1. ^ Stein 1993, Chapter 6
  2. ^ Atiyah & Singer 1968, p. 486

References[edit]

Further reading[edit]

Here are some of the standard reference books

External links[edit]