Symbol of a differential operator

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.


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