This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative.
Assume that there is a map from a function space to another function space and a function so that is the image of i.e.,.
A differential operator is represented as a linear combination, finitely generated by and its derivatives containing higher degree such as
where the list of non-negative integers is called a multi-index, is called the length of , are functions on some open domain in n-dimensional space, and .
The derivative above is one as functions or, sometimes, distributions or hyperfunctions and or sometimes, .
The most common differential operator is the action of taking the derivative. Common notations for taking the first derivative with respect to a variable x include:
, , and .
When taking higher, nth order derivatives, the operator may be written:
, , , or .
The derivative of a function f of an argumentx is sometimes given as either of the following:
The D notation's use and creation is credited to Oliver Heaviside, who considered differential operators of the form
In writing, following common mathematical convention, the argument of a differential operator is usually placed on the right side of the operator itself. Sometimes an alternative notation is used: The result of applying the operator to the function on the left side of the operator and on the right side of the operator, and the difference obtained when applying the differential operator to the functions on both sides, are denoted by arrows as follows:
Such a bidirectional-arrow notation is frequently used for describing the probability current of quantum mechanics.
where the line over f(x) denotes the complex conjugate of f(x). If one moreover adds the condition that f or g vanishes as 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 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 a 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.
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. For example we have 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.
where jk: Γ(E) → Γ(Jk(E)) is the prolongation that associates to any section of E its k-jet.
This just means that for a given sections of E, the value of P(s) at a point x ∈ M 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 (linear) local operator is differential.
Relation to commutative algebra
An equivalent, but purely algebraic description of linear differential operators is as follows: an R-linear map P is a kth-order linear differential operator, if for any k + 1 smooth functions we have