Knowlpedia

Definition

Principal symbol of a differential operator

The homogeneous bundle map on the cotangent bundle obtained from the highest-order part of a differential operator.

Definition

Let D:Γ∞(E)→Γ∞(F)D:\Gamma^\infty(E)\to\Gamma^\infty(F) be a of order mm. Its principal symbol is the

σm(D):π∗E⟶π∗F\sigma_m(D):\pi^*E\longrightarrow\pi^*F

over the T∗MT^*M, homogeneous of degree mm in each covector. In local coordinates and frames, if

D=∑∣α∣≤maα(x)∂α,D=\sum_{|\alpha|\leq m}a_\alpha(x)\partial^\alpha,

then

σm(D)(x,ξ)=∑∣α∣=maα(x)ξα:Ex→Fx.\sigma_m(D)(x,\xi)=\sum_{|\alpha|=m}a_\alpha(x)\xi^\alpha:E_x\to F_x.

Although this formula uses coordinates, its highest-order transformation law makes σm(D)\sigma_m(D) intrinsic.

Commutator characterization

The principal symbol can be recovered without coordinates. Repeated commutators of DD with multiplication operators are order zero, and their value at xx depends only on the differentials of the functions there. Polarizing this symmetric mm-linear expression yields σm(D)(x,ξ)\sigma_m(D)(x,\xi). Consequently, adding an operator of order at most m−1m-1 does not change the principal symbol.

Algebraic properties and ellipticity

For operators of orders mm and nn,

σm+n(D2D1)=σn(D2)σm(D1).\sigma_{m+n}(D_2D_1)=\sigma_n(D_2)\sigma_m(D_1).

Thus leading symbols turn composition of differential operators into pointwise composition of homogeneous bundle maps. An operator is precisely when this symbol is invertible at every nonzero covector. See Lawson and Michelsohn, chapter III, §1 for the symbol calculus used in spin geometry.

Conventions and scope
References
  1. H. B. Lawson Jr. and M.-L. Michelsohn, Spin Geometry, Princeton University Press, 1989. Publisher record. Relevant: chapter III, §1, symbols and ellipticity.
  2. N. Berline, E. Getzler, and M. Vergne, Heat Kernels and Dirac Operators, Springer, 1992. Publisher record. Relevant: chapter 2, differential operators and symbols.