symbol of a differential operator



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

σ(D):T *X XEE \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 xXx \in X on a cotangent vector of the form (df) xΓ(T *X) x(\mathbf{d}f)_x \in \Gamma(T^* X)_x by

σ(D) x:df x[D,f] x, \sigma(D)_x \;\colon\; \mathbf{d}f_x \mapsto [D,f]_x \,,

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

The symbol may naturally be thought of as an element in the K-theory of XX (Freed).



(symbol map)

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


(bicharacteristic flow)

For EE 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.


For instance chapter 2.5 of

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

  • Dan Freed, Geometry of Dirac operators (pdf)

Revised on August 4, 2017 15:19:41 by Urs Schreiber (