pseudo-differential operator

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

Pseudo-differential Operator

References

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