In
mathematics, the symbol of a linear differential operator is a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
representing a
differential operator, which is obtained, roughly speaking, by 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
In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory, e.g. in m ...
. 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
Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry.
The following phenomena are described as ''hyperbolic'' because they ...
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
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
R
''d''. Then ''P'' is a polynomial in the derivative ''D'', which in
multi-index
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices. ...
notation can be written
:
The total symbol of ''P'' is the polynomial ''p'':
:
The leading symbol, also known as the principal symbol, is the highest-degree component of ''p'' :
:
and is of importance later because it is the only part of the symbol that transforms as a
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other 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,
:
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 operator
In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory, e.g. in m ...
s.
Vector bundles
Let ''E'' and ''F'' be
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
s over a
closed manifold ''X'', and suppose
:
is a differential operator of order
. In
local coordinates
Local coordinates are the ones used in a ''local coordinate system'' or a ''local coordinate space''. Simple examples:
* Houses. In order to work in a house construction, the measurements are referred to a control arbitrary point that will allow ...
on ''X'', we have
:
where, for each
multi-index
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices. ...
α,
is a
bundle map
In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. There ...
, symmetric on the indices α.
The ''k''
th order coefficients of ''P'' transform as a
symmetric tensor
In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments:
:T(v_1,v_2,\ldots,v_r) = T(v_,v_,\ldots,v_)
for every permutation ''σ'' of the symbols Alternatively, a symmetric tensor of orde ...
:
from the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
of the ''k''
th symmetric power In mathematics, the ''n''-th symmetric power of an object ''X'' is the quotient of the ''n''-fold product X^n:=X \times \cdots \times X by the permutation action of the symmetric group \mathfrak_n.
More precisely, the notion exists at least in the ...
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 ''x''
''i'' permits a local trivialization of the cotangent bundle by the coordinate differentials d''x''
''i'', 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
:
on each section ''u'' of ''E''. Here ''P''
νμ is the scalar differential operator defined by
:
With this trivialization, the principal symbol can now be written
:
In the cotangent space over a fixed point ''x'' of ''X'', the symbol
defines a
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
of degree ''k'' in
with values in
.
The differential operator
is
elliptic
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
if its symbol is invertible; that is for each nonzero
the bundle map
is invertible. On a
compact manifold
In mathematics, a closed manifold is a manifold without boundary that is compact.
In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components.
Examples
The only connected one-dimensional example is ...
, it follows from the elliptic theory that ''P'' is a
Fredholm operator
In mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operator ''T'' : '' ...
: it has finite-dimensional
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learn ...
and cokernel.
See also
*
Multiplier (Fourier analysis)
In Fourier analysis, a multiplier operator is a type of linear operator, or transformation of functions. These operators act on a function by altering its Fourier transform. Specifically they multiply the Fourier transform of a function by a speci ...
*
Atiyah–Singer index theorem (section on symbol of operator)
References
*
*.
* {{citation, last=Wells, first=R.O., authorlink=Raymond O. Wells, Jr., title=Differential analysis on complex manifolds, year=1973, publisher=Springer-Verlag, isbn=0-387-90419-0.
Differential operators
Vector bundles