Symbol of a differential operator

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In mathematics, differential operators have symbols, which are roughly speaking the algebraic part of the terms involving the most derivatives.

Formal definition

Let ${\displaystyle E_{1},E_{2}}$  be vector bundles over a closed manifold X, and suppose

${\displaystyle P:C^{\infty }(E_{1})\to C^{\infty }(E_{2})}$ 

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

${\displaystyle Pu=a^{i_{1}i_{2}\dots i_{k}}{\frac {\partial ^{k}u}{\partial x^{i_{1}}\,\partial x^{i_{2}}\cdots \partial x^{i_{k}}}}+{\text{lower order terms}}}$ 

where

${\displaystyle a^{i_{1}\dots i_{k}}}$ 

is a bundle map

${\displaystyle E_{1}\to E_{2}}$ 

depending symmetrically on the ${\displaystyle i_{j}}$ , and we sum over the indices ${\displaystyle i_{j}}$ . This top order piece transforms as a symmetric tensor under change of coordinates, so it defines the symbol:

${\displaystyle \sigma (P):S^{k}(T^{*}X)\otimes E_{1}\to E_{2}}$ .

View the symbol ${\displaystyle \sigma (P)}$  as a homogeneous polynomial of degree ${\displaystyle k}$  in ${\displaystyle T^{*}X}$  with values in ${\displaystyle Hom(E_{1},E_{2})}$ .

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. It follows from the elliptic theory that ${\displaystyle P}$  has finite dimensional kernel and cokernel.

References

• Daniel S. Freed Geometry of Dirac operators. p.8.