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.
The most commonly used differential operator is the action of taking the derivative itself. Common notations for this operator include:
First derivatives are signified as above, but when taking higher, nth derivatives, the following alterations are useful:
For a function f of an argument x, the derivative operator is sometimes given:
The D notation's use and creation is credited to Oliver Heaviside, who considered differential operators of the form
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, or theta 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
Given a linear differential operator T
the adjoint of this operator is defined as the operator T * such that
In the functional space of square integrable functions, the scalar product is defined by
If one moreover adds the condition that f or g vanishes for and , one can also define the adjoint of T by
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 T * 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.
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.
The Sturm-Liouville operator is a well-known example of formal self-adjoint operator. This second-order linear differential operator L can be written in the form
This property can be proven using the formal adjoint definition above.
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.
In differential geometry and algebraic geometry it is often convenient to have a coordinate-independent description of differential operators between two vector bundles. Let E and F be two vector bundles over a manifold M. An -linear mapping of sections is said to be a kth-order linear differential operator if it factors through the jet bundle . In other words, there exists a linear mapping of vector bundles
where denotes the map induced by on sections , and is the canonical (or universal) kth-order differential operator.
This just means that for a given sections s of E, the value of P(s) at a point is fully determined by the kth-order infinitesimal behavior of s in x. In particular this implies that P(s)(x) is determined by the germ of s in x, 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 P is a kth-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.