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Differential operator
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In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another (in the style of a higher-order function in computer science).
There are certainly reasons not to restrict to linear operators; for instance the Schwarzian derivative is a well-known non-linear operator.
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In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another (in the style of a higher-order function in computer science).
There are certainly reasons not to restrict to linear operators; for instance the Schwarzian derivative is a well-known non-linear operator. Only the linear case will be addressed here.
Notations
The most commonly used differential operator is the action of taking the derivative itself. Common notations for this operator include:
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- where the variable with respect to which one is differentiating is clear, and
- where the variable is declared explicitly.
First derivatives are signified as above, but when taking higher, nth derivatives, the following alterations are useful:
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The D notation's use and creation is credited to Oliver Heaviside, who considered differential operators of the form
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in his study of differential equations.
One of the most frequently seen differential operators is the Laplacian operator, defined by
Another differential operator is the Θ operator, defined by
This is sometimes also called the homogeneity operator, because its eigenfunctions are the monomials in z:
In n variables the homogeneity operator is given by
As in one variable, the eigenspaces of Θ are the spaces of homogeneous polynomials.
Adjoint of an operator
Given a linear differential operator T
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the adjoint of this operator is defined as the operator such that
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where the notation is used for the scalar product or inner product. This definition therefore depends on the definition of the scalar product.
Formal adjoint in one variable
In the functional space of square integrable functions, the scalar product is defined by
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If one moreover adds the condition that f or g vanishes for and , one can also define the adjoint of T by
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This formula does not explicitly depend on the definition of the scalar product. It is therefore sometimes chosen as a definition of the adjoint operator. When is defined according to this formula, it is called the formal adjoint of T.
A (formally) self-adjoint operator is an operator equal to its own (formal) adjoint.
Several variables
If Ω is a domain in Rn, and P a differential operator on Ω, then the adjoint of P is defined in L2(Ω) by duality in the analogous manner:
for all smooth L2 functions f, g. Since smooth functions are dense in L2, this defines the adjoint on a dense subset of L2: P* is a densely-defined operator.
Example
The Sturm-Liouville operator is a well-known example of formal self-adjoint operator. This second order linear differential operators L can be written in the form
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This property can be proven using the formal adjoint definition above.
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This operator is central to Sturm-Liouville theory where the eigenfunctions (analogues to eigenvectors) of this operator are considered.
Properties of differential operators
Differentiation is linear, i.e.,
where f and g are functions, and a is a constant.
Any polynomial in D with function coefficients is also a differential operator. We may also compose differential operators by the rule
Some care is then required: firstly any function coefficients in the operator D2 must be differentiable as many times as the application of D1 requires. To get a ring of such operators we must assume derivatives of all orders of the coefficients used. Secondly, this ring will not be commutative: an operator gD isn't the same in general as Dg. In fact we have for example the relation basic in quantum mechanics:
The subring of operators that are polynomials in D with constant coefficients is, by contrast, commutative. It can be characterised another way: it consists of the translation-invariant operators.
The differential operators also obey the shift theorem.
Several variables
The same constructions can be carried out with partial derivatives, differentiation with respect to different variables giving rise to operators that commute (see symmetry of second derivatives).
Coordinate-independent description and relation to commutative algebra
In differential geometry and algebraic geometry it is often convenient to have a coordinate-independent description of differential operators between two vector bundles. Let and be two vector bundles over a manifold . An -linear mapping of sections is said to be a k-th order linear differential operator if it factors through the jet bundle .
In other words, there exists a linear mapping of vector bundles
such that
where denotes the map induced by on sections , and is the canonical (or universal) k-th order differential operator.
This just means that for a given sections of , the value of at a point is fully determined by the k-th order infinitesimal behavior of in . In particular this implies that is determined by the germ of in , which is expressed by saying that differential operators are local. A foundational result is the Peetre theorem showing that the converse is also true: any local operator is differential.
An equivalent, but purely algebraic description of linear differential operators is as follows: an -linear map is a k-th order linear differential operator, if for any k+1 smooth functions we have
Here the bracket is defined as the commutator
This characterization of linear differential operators shows that they are particular mappings between modules over a commutative algebra, allowing the concept to be seen as a part of commutative algebra.
Examples
See also
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