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#redirect [[Differential operator]]
{{Short description|Representation of differential polynomial operators}}
{{No footnotes|date=May 2020}}
{{mergeto|Differential operator|discuss=Talk:Differential_operator#Merge_with_Symbol_of_a_differential_operator|date=March 2023}}
In [[mathematics]], the '''symbol of a linear differential operator''' is a [[polynomial]] representing a [[differential operator]], which is obtained, roughly speaking, by replacing each [[partial derivative]] by a new variable. The symbol of a differential operator has broad applications to [[Fourier analysis]]. In particular, in this connection it leads to the notion of a [[pseudo-differential operator]]. The highest-order terms of the symbol, known as the principal symbol, almost completely controls the qualitative behavior of solutions of a [[partial differential equation]]. Linear [[elliptic partial differential equation]]s can be characterized as those whose principal symbol is nowhere zero. In the study of [[hyperbolic partial differential equation|hyperbolic]] and [[parabolic partial differential equation]]s, zeros of the principal symbol correspond to the [[method of characteristics|characteristics]] of the partial differential equation. Consequently, the symbol is often fundamental for the solution of such equations, and is one of the main computational devices used to study their singularities.
==Definition==
===Operators on Euclidean space===
Let ''P'' be a linear differential operator of order ''k'' on the [[Euclidean space]] '''R'''<sup>''d''</sup>. Then ''P'' is a polynomial in the derivative ''D'' — a polynomial whose coefficients are real-valued functions defined on '''R'''<sup>''d''</sup>. In [[multi-index]] notation, this polynomial can be written as
:<math>P = p(x,D) = \sum_{|\alpha|\le k} a_\alpha(x) D^\alpha.</math>
The '''total symbol''' of ''P'' is the polynomial ''p'':
:<math> p(x,\xi) = \sum_{|\alpha|\le k} a_\alpha(x)\xi^\alpha.</math>
The '''leading symbol''', also known as the '''principal symbol''', is the highest-degree component of ''p'' :
:<math>\sigma_P (\xi) = \sum_{|\alpha|= k} a_\alpha\xi^\alpha</math>
and is of importance later because it is the only part of the symbol that transforms as a [[tensor]] under changes to the coordinate system.
The symbol of ''P'' appears naturally in connection with the [[Fourier transform]] as follows. Let ƒ be a [[Schwartz function]]. Then by the inverse Fourier transform,
:<math>Pf(x) = \frac{1}{(2\pi)^{\frac{d}{2}}} \int\limits_{\mathbf{R}^d} e^{ ix\cdot\xi} p(x,i\xi)\hat{f}(\xi)\, d\xi.</math>
This exhibits ''P'' as a [[Fourier multiplier]]. A more general class of functions ''p''(''x'',ξ) which satisfy at most polynomial growth conditions in ξ under which this integral is well-behaved comprises the [[pseudo-differential operator]]s.
===Vector bundles===
Let ''E'' and ''F'' be [[vector bundle]]s over a [[closed manifold]] ''X'', and suppose
:<math> P: C^\infty(E) \to C^\infty(F) </math>
is a differential operator of order <math> k </math>. In [[local coordinates]] on ''X'', we have
:<math> Pu(x) = \sum_{|\alpha| = k} P^\alpha(x) \frac {\partial^\alpha u} {\partial x^{\alpha}} + \text{lower-order terms}</math>
where, for each [[multi-index]] α, <math> P^\alpha(x):E \to F</math> is a [[bundle map]], symmetric on the indices α.
The ''k''<sup>th</sup> order coefficients of ''P'' transform as a [[symmetric tensor]]
:<math> \sigma_P: S^k (T^*X) \otimes E \to F </math>
whose ___domain is the [[tensor product]] of the ''k''<sup>th</sup> [[symmetric power]] of the [[cotangent bundle]] of ''X'' with ''E'', and whose codomain is ''F''. This symmetric tensor is known as the '''principal symbol''' (or just the '''symbol''') of ''P''.
The coordinate system ''x''<sup>''i''</sup> permits a local trivialization of the cotangent bundle by the coordinate differentials d''x''<sup>''i''</sup>, which determine fiber coordinates ξ<sub>''i''</sub>. In terms of a basis of frames ''e''<sub>μ</sub>, ''f''<sub>ν</sub> of ''E'' and ''F'', respectively, the differential operator ''P'' decomposes into components
:<math>(Pu)_\nu = \sum_\mu P_{\nu\mu}u_\mu</math>
on each section ''u'' of ''E''. Here ''P''<sub>νμ</sub> is the scalar differential operator defined by
:<math>P_{\nu\mu} = \sum_{\alpha} P_{\nu\mu}^\alpha\frac{\partial}{\partial x^\alpha}.</math>
With this trivialization, the principal symbol can now be written
:<math>(\sigma_P(\xi)u)_\nu = \sum_{|\alpha|=k} \sum_{\mu}P_{\nu\mu}^\alpha(x)\xi_\alpha u^\mu.</math>
In the cotangent space over a fixed point ''x'' of ''X'', the symbol <math> \sigma_P </math> defines a [[homogeneous polynomial]] of degree ''k'' in <math> T^*_x X </math> with values in <math> \operatorname{Hom}(E_x, F_x) </math>.
The differential operator <math> P </math> is [[elliptic differential operator|elliptic]] if its symbol is invertible; that is for each nonzero <math> \theta \in T^*X </math> the bundle map <math> \sigma_P (\theta, \dots, \theta)</math> is invertible. On a [[compact manifold]], it follows from the elliptic theory that ''P'' is a [[Fredholm operator]]: it has finite-dimensional [[kernel (algebra)|kernel]] and cokernel.
==See also==
* [[Multiplier (Fourier analysis)]]
*[[Atiyah–Singer index theorem|Atiyah–Singer index theorem (section on symbol of operator)]]
==References==
{{reflist}}
* {{citation|first=Daniel S.|last=Freed|title=Geometry of Dirac operators|page=8 |date=1987 |citeseerx=10.1.1.186.8445 }}
*{{citation|mr=0717035|first=L.|last= Hörmander|authorlink=Lars Hörmander|title=The analysis of linear partial differential operators I|series= Grundl. Math. Wissenschaft. |volume= 256 |publisher=Springer |year=1983|isbn=3-540-12104-8 |doi=10.1007/978-3-642-96750-4}}.
* {{citation|last=Wells|first=R.O.|authorlink=Raymond O. Wells, Jr.|title=Differential analysis on complex manifolds|year=1973|publisher=Springer-Verlag|isbn=0-387-90419-0}}.
[[Category:Differential operators]]
[[Category:Vector bundles]]
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