# Differential operator

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Differential operator

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).

This article considers only linear operators, even though the Schwarzian derivative is a prominent example of a non-linear operator.

## Notations

The most commonly used differential operator is the action of taking the derivative itself. Common notations for this operator include: ${d \over dx}$ $D,\,$ where the variable with respect to which one is differentiating is clear, and $D_x,\,$ where the variable is declared explicitly. $\partial_x$, is an alternative notation.

First derivatives are signified as above, but when taking higher, nth derivatives, the following alterations are useful: $d^n \over dx^n$ $D^n\,$ $D^n_x.\,$

For a function f of an argument x, the derivative operator is sometimes given as either of the following: $[f(x)]'\,\!$ $f'(x)\,\!$

The D notation's use and creation is credited to Oliver Heaviside, who considered differential operators of the form $\sum_{k=0}^n c_k D^k$

in his study of differential equations.

One of the most frequently seen differential operators is the Laplacian operator, defined by $\Delta=\nabla^{2}=\sum_{k=1}^n {\partial^2\over \partial x_k^2}.$

Another differential operator is the Θ operator, or theta operator, defined by $\Theta = z {d \over dz}.$

This is sometimes also called the homogeneity operator, because its eigenfunctions are the monomials in z: $\Theta (z^k) = k z^k,\quad k=0,1,2,\dots$

In n variables the homogeneity operator is given by $\Theta = \sum_{k=1}^n x_k \frac{\partial}{\partial x_k}.$

As in one variable, the eigenspaces of Θ are the spaces of homogeneous polynomials.

Given a linear differential operator T $Tu = \sum_{k=0}^n a_k(x) D^k u$

the adjoint of this operator is defined as the operator T * such that $\langle Tu,v \rangle = \langle u, T^*v \rangle$

where the notation $\langle\cdot,\cdot\rangle$ 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 $\langle f, g \rangle = \int_a^b f(x) \, \overline{g(x)} \,dx.$

If one moreover adds the condition that f or g vanishes for $x \to a$ and $x \to b$, one can also define the adjoint of T by $T^*u = \sum_{k=0}^n (-1)^k D^k [a_k(x)u].\,$

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.

### 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: $\langle f, P^* g\rangle_{L^2(\Omega)} = \langle P f, g\rangle_{L^2(\Omega)}$

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 operator L can be written in the form $Lu = -(pu')'+qu=-(pu''+p'u')+qu=-pu''-p'u'+qu=(-p) D^2 u +(-p') D u + (q)u.\;\!$

This property can be proven using the formal adjoint definition above. \begin{align} L^*u & {} = (-1)^2 D^2 [(-p)u] + (-1)^1 D [(-p')u] + (-1)^0 (qu) \\ & {} = -D^2(pu) + D(p'u)+qu \\ & {} = -(pu)''+(p'u)'+qu \\ & {} = -p''u-2p'u'-pu''+p''u+p'u'+qu \\ & {} = -p'u'-pu''+qu \\ & {} = -(pu')'+qu \\ & {} = Lu \end{align}

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., $D(f+g) = (Df)+(Dg)\,$ $D(af) = a(Df)\,$

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 $(D_1 \circ D_2)(f) = D_1(D_2(f)).\,$

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: $Dx - xD = 1.\,$

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

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 differentiable manifold M. An R-linear mapping of sections P : Γ(E) → Γ(F) is said to be a kth-order linear differential operator if it factors through the jet bundle Jk(E). In other words, there exists a linear mapping of vector bundles $i_P: J^k(E) \rightarrow F\,$

such that $P = i_P\circ j^k$

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 s 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 $f_0,\ldots,f_k \in C^\infty(M)$ we have $[f_k,[f_{k-1},[\cdots[f_0,P]\cdots]]=0.$

Here the bracket $[f,P]:\Gamma(E)\rightarrow \Gamma(F)$ is defined as the commutator $[f,P](s)=P(f\cdot s)-f\cdot P(s).\,$

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

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