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.

[edit] Formal definition

Let $E 1, E 2$ be vector bundles over a closed manifold X, and suppose

$P: C^\infty(E_1) \to C^\infty(E_2)$

is a differential operator of order $k$ . In local coordinates we have

$Pu = a^{i_1 i_2 \dots i_k} \frac {\partial^k u} {\partial x^{i_1}\, \partial x^{i_2} \cdots \partial x^{i_k}} + \text{lower order terms}$

where

$a^{i_1 \dots i_k}$

is a bundle map

$E_{1} \to E_{2}$

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

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

View the symbol $σ(P)$ as a homogeneous polynomial of degree $k$ in $T * X$ with values in $\operatorname{Hom}(E_1, E_2)$ .

The differential operator $P$ is elliptic if its symbol is invertible; that is for each nonzero $\theta \in T^*X$ the bundle map $\sigma(P) (\theta, \dots, \theta)$ is invertible. It follows from the elliptic theory that $P$ has finite dimensional kernel and cokernel.