Symbol of a differential operator

In mathematics, the symbol of a linear differential operator associates to a differential operator a polynomial by, roughly speaking, replacing each partial derivative by a new variable. The symbol of a differential operator has broad applications to Fourier analysis. In particular, in this connection it leads to the notion of a pseudo-differential operator. The highest-order terms of the symbol, known as the principal symbol, almost completely controls the qualitative behavior of solutions of a partial differential equation. Linear elliptic partial differential equations can be characterized as those whose principal symbol is nowhere zero. In the study of hyperbolic and parabolic partial differential equations, zeros of the principal symbol correspond to the characteristics of the partial differential equation. Consequently, the symbol is often fundamental for the solution of such equations, and is one of the main computational devices used to study their singularities.

Definition

Operators on Euclidean space

Let P be a linear differential operator of order k on the Euclidean space R^d. Then P is a polynomial in the derivative D, which in multi-index notation can be written

$P=p(x,D)=\sum _{{|\alpha |\leq k}}a_{\alpha }(x)D^{\alpha }.$

The total symbol of P is the polynomial p:

$\sigma _{P}(\xi )=p(x,\xi )=\sum _{{|\alpha |\leq k}}a_{\alpha }(x)\xi ^{\alpha }.$

The leading symbol, also known as the principal symbol, is the highest degree component of σ_P :

$\sigma _{P}(\xi )=\sum _{{|\alpha |=k}}a_{\alpha }\xi ^{\alpha }$

and is of importance later because it is the only part of the symbol that transforms as a tensor under changes to the coordinate system.

The symbol of P appears naturally in connection with the Fourier transform as follows. Let ƒ be a Schwartz function. Then by the inverse Fourier transform,

$Pf(x)={\frac {1}{2\pi }}\int _{{{\mathbf {R}}^{d}}}e^{{ix\cdot \xi }}p(x,i\xi ){\hat {f}}(\xi )\,d\xi .$

This exhibits P as a Fourier multiplier. A more general class of functions p(x,ξ) which satisfy at most polynomial growth conditions in ξ under which this integral is well-behaved comprises the pseudo-differential operators.

Vector bundles

Let E and F be vector bundles over a closed manifold X, and suppose

$P:C^{\infty }(E)\to C^{\infty }(F)$

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

$Pu(x)=\sum _{{|\alpha |=k}}P^{\alpha }(x){\frac {\partial ^{\alpha }u}{\partial x^{{\alpha }}}}+{\text{lower order terms}}$

where, for each multi-index α, $P^{\alpha }(x):E\to F$ is a bundle map, symmetric on the indices α.

The k^th order coefficients of P transform as a symmetric tensor

$\sigma _{P}:S^{{k}}(T^{*}X)\otimes E\to F$

from the tensor product of the k^th symmetric power of the cotangent bundle of X with E to F. This symmetric tensor is known as the principal symbol (or just the symbol) of P.

The coordinate system xⁱ permits a local trivialization of the cotangent bundle by the coordinate differentials dxⁱ, which determine fiber coordinates ξ_i. In terms of a basis of frames e_μ, f_ν of E and F, respectively, the differential operator P decomposes into components

$(Pu)_{\nu }=\sum _{\mu }P_{{\nu \mu }}u_{\mu }$

on each section u of E. Here P_νμ is the scalar differential operator defined by

$P_{{\nu \mu }}=\sum _{{\alpha }}P_{{\nu \mu }}^{\alpha }{\frac {\partial }{\partial x^{\alpha }}}.$

With this trivialization, the principal symbol can now be written

$(\sigma _{P}(\xi )u)_{\nu }=\sum _{{|\alpha |=k}}\sum _{{\mu }}P_{{\nu \mu }}^{\alpha }(x)\xi _{\alpha }u^{\mu }.$

In the cotangent space over a fixed point x of X, the symbol $\sigma _{P}$ defines a homogeneous polynomial of degree k in $T_{x}^{*}X$ with values in $\operatorname {Hom}(E_{x},F_{x})$.

The differential operator $P$ is elliptic if its symbol is invertible; that is for each nonzero $\theta \in T^{*}X$ the bundle map $\sigma _{P}(\theta ,\dots ,\theta )$ is invertible. On a compact manifold, it follows from the elliptic theory that P is a Fredholm operator: it has finite-dimensional kernel and cokernel.

See also

• Multiplier (Fourier analysis)
• Atiyah-Singer index theorem (section on symbol of operator)

References

• Freed, Daniel S., Geometry of Dirac operators, p. 8
• Hörmander, L. (1983), The analysis of linear partial differential operators I, Grundl. Math. Wissenschaft., 256, Springer, ISBN 3-540-12104-8, MR 0717035.
• Wells, R.O. (1973), Differential analysis on complex manifolds, Springer-Verlag, ISBN 0-387-90419-0.