Symbol of a differential operator

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

Formal definition

Let $E\_1,\; E\_2$ be vector bundles over a closed manifold "X", and suppose

:$P:\; C^infty(E\_1)\; o\; C^infty(E\_2)$

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

:$Pu\; =\; a^\{i\_1\; i\_2\; dots\; i\_k\}\; frac\; \{partial^k\; u\}\; \{partial\; x^\{i\_1\},\; partial\; x^\{i\_2\}\; cdots\; partial\; x^\{i\_k\; +\; ext\{lower\; order\; terms\}$

where all $a^\{i\_1\; dots\; i\_k\}:E\_\{1\}\; o\; E\_\{2\}$ are bundle mapsdepending symmetrically on the $i\_j$ and where 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:

:$sigma(P):\; S^\{k\}\; (T^*X)\; otimes\; E\_\{1\}\; o\; E\_\{2\}.$

View the symbol $sigma(P)$ 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 $heta\; in\; T^*X$ the bundle map $sigma(P)\; (\; heta,\; dots,\; heta)$ is invertible. It follows from the elliptic theory that $P$ has finite dimensional kernel and cokernel.

ee also

* Multiplier (Fourier analysis)

References

*Daniel S. Freed "Geometry of Dirac operators." p.8.