Symbol of a differential operator

Let P be a linear differential operator of order k on the Euclidean space R^d. Then P is a polynomial in the derivative D, which in multi-index notation can be written

${\displaystyle P=p(x,D)=\sum _{|\alpha |\leq k}a_{\alpha }(x)D^{\alpha }.}$

The total symbol of P is the polynomial p:

${\displaystyle p(x,\xi )=\sum _{|\alpha |\leq k}a_{\alpha }(x)\xi ^{\alpha }.}$

The leading symbol, also known as the principal symbol, is the highest-degree component of p :

${\displaystyle \sigma _{P}(\xi )=\sum _{|\alpha |=k}a_{\alpha }\xi ^{\alpha }}$

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,

${\displaystyle Pf(x)={\frac {1}{(2\pi )^{d}}}\int _{\mathbf {R} ^{d}}e^{ix\cdot \xi }p(x,i\xi ){\hat {f}}(\xi )\,d\xi .}$

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 operators.

Let E and F be vector bundles over a closed manifold X, and suppose

${\displaystyle P:C^{\infty }(E)\to C^{\infty }(F)}$

is a differential operator of order ${\displaystyle k}$. In local coordinates on X, we have

${\displaystyle Pu(x)=\sum _{|\alpha |=k}P^{\alpha }(x){\frac {\partial ^{\alpha }u}{\partial x^{\alpha }}}+{\text{lower-order terms}}}$

where, for each multi-index α, ${\displaystyle P^{\alpha }(x):E\to F}$ is a bundle map, symmetric on the indices α.

The k^th order coefficients of P transform as a symmetric tensor

${\displaystyle \sigma _{P}:S^{k}(T^{*}X)\otimes E\to F}$

from the tensor product of the k^th symmetric power of the cotangent bundle of X with E to F. This symmetric tensor is known as the principal symbol (or just the symbol) of P.

The coordinate system xⁱ permits a local trivialization of the cotangent bundle by the coordinate differentials dxⁱ, which determine fiber coordinates ξ_i. In terms of a basis of frames e_μ, f_ν of E and F, respectively, the differential operator P decomposes into components

${\displaystyle (Pu)_{\nu }=\sum _{\mu }P_{\nu \mu }u_{\mu }}$

on each section u of E. Here P_νμ is the scalar differential operator defined by

${\displaystyle P_{\nu \mu }=\sum _{\alpha }P_{\nu \mu }^{\alpha }{\frac {\partial }{\partial x^{\alpha }}}.}$

With this trivialization, the principal symbol can now be written

${\displaystyle (\sigma _{P}(\xi )u)_{\nu }=\sum _{|\alpha |=k}\sum _{\mu }P_{\nu \mu }^{\alpha }(x)\xi _{\alpha }u^{\mu }.}$

In the cotangent space over a fixed point x of X, the symbol ${\displaystyle \sigma _{P}}$ defines a homogeneous polynomial of degree k in ${\displaystyle T_{x}^{*}X}$ with values in ${\displaystyle \operatorname {Hom} (E_{x},F_{x})}$.

The differential operator ${\displaystyle P}$ is elliptic if its symbol is invertible; that is for each nonzero ${\displaystyle \theta \in T^{*}X}$ the bundle map ${\displaystyle \sigma _{P}(\theta ,\dots ,\theta )}$ is invertible. On a compact manifold, it follows from the elliptic theory that P is a Fredholm operator: it has finite-dimensional kernel and cokernel.