nLab symbol of a differential operator

Definition

For $X$ a smooth manifold, $E \to X$ a vector bundle and $D : \Gamma(E) \to \Gamma(E)$ a linear differential operator on sections of $E$, its symbol is the bundle morphism

$$\sigma(D) \;:\; T^* X \otimes_X E \to E$$

from the tensor product of vector bundles with the cotangent bundle which is given at any point $x \in X$ on a cotangent vector of the form $(\mathbf{d}f)_x \in \Gamma(T^* X)_x$ by

$$\sigma(D)_x \;\colon\; \mathbf{d}f_x \mapsto [D,f]_x \,,$$

where in the commutator on the right we regard multiplication by $f$ as an endomorphism of $\Gamma(E)$.

The symbol of an elliptic differential operator may naturally be thought of as an element in the topological K-theory of $T^\ast X$ (see Freed 1987, p. 13).

Generally, the symbol of a possibly non-linear differential operator is similarly the map on the cotangent bundle given by “replacing partial derivatives by covectors” in the definition of the differential operator. In this case the principal symbol is the highest degree homogeneous component of the symbol. Many authors say total symbol for symbol in the sense of this entry and symbol for a principal symbol.

Examples

Definition

(symbol map)

For $X = \mathbb{R}^n$ a Cartesian space and $D$ the Dirac operator of the flat connection, the symbol of $D$ reproduces the symbol map between differential forms and Clifford algebra elements.

Example

(bicharacteristic flow)

For $E$ a real trivial line bundle then the principal symbol is equivalently just a real-valued smooth function on the cotangent bundle. Since any cotangent bundle is canonically a symplectic manifold, in this case the symbol may be regarded as a Hamiltonian funtion. The corresponding Hamiltonian flow is called the bicharacteristic flow of the given differential operator.

Related concepts

• symbol order, symbol map, principal symbol

• elliptic differential operator, elliptic chain complex

• hyperbolic differential operator

• propagation of singularities theorem

References

• Nigel Higson, John Roe, chapter 2.5 of Lectures on operator K-theory and the Atiyah-Singer Index Theorem (pdf)

• Daniel Freed, Geometry of Dirac operators (1987) [pdf, pdf]

See also

• Wikipedia, Symbol of a differential operator