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In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator to include non-integer orders of differentiation (see Differintegral). Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory.
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
- p(x, D)
for a polynomial p in D 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 an algebraic problem involving their symbols, and this is the essence of micro-local analysis.
References
Here are some of the standard reference books
- Michael E. Taylor, Pseudodifferential Operators, Princeton Univ. Press 1981. ISBN 0691082820
- M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag 2001. ISBN 354041195X
- Francois Treves, Introduction to Pseudo Differential and Fourier Integral Operators, (University Series in Mathematics), Plenum Publ. Co. 1981. ISBN 0306404044
- F. G. Friedlander and M. Joshi, Introduction to the Theory of Distributions, Cambridge University Press 1999. ISBN 0521649714
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