Differential operator Information & Differential operator Links at HealthHaven.com

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.

[edit] 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.

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:

$[f(x)]'\,\!$

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^[1]

$\Theta = z {d \over dz}.$

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.

[edit] Adjoint of an operator

[edit] 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.

[edit] 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.

[edit] 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.

[edit] 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.

[edit] 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).

[edit] 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 $E$ and $F$ be two vector bundles over a manifold $M$ . An $\mathbb{R}$ -linear mapping of sections $P: \Gamma(E) \rightarrow \Gamma(F)\,$ is said to be a kth-order linear differential operator if it factors through the jet bundle $J^k(E)\,$ . In other words, there exists a linear mapping of vector bundles

$i_P: J^k(E) \rightarrow F\,$

such that

$P = \hat{i}_P\circ j^k$

where $\hat{i}_P$ denotes the map induced by $i_P\,$ on sections , and $j^k:\Gamma(E)\rightarrow \Gamma(J^k(E))\,$ 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 $x\in M$ 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 local operator is differential.

An equivalent, but purely algebraic description of linear differential operators is as follows: an $\mathbb{R}$ -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.

[edit] Examples

In applications to the physical sciences, operators such as the Laplace operator play a major role in setting up and solving partial differential equations.

In differential topology the exterior derivative and Lie derivative operators have intrinsic meaning.

In abstract algebra, the concept of a derivation allows for generalizations of differential operators which do not require the use of calculus. Frequently such generalizations are employed in algebraic geometry and commutative algebra. See also jet (mathematics).

[edit] See also

[edit] References

^ E. W. Weisstein. "Theta Operator". http://mathworld.wolfram.com/ThetaOperator.html. Retrieved 2009-06-12.

[0] E. W. Weisstein. "Theta Operator". http://mathworld.wolfram.com/ThetaOperator.html. Retrieved 2009-06-12.

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Contents

[edit] Notations

[edit] Adjoint of an operator

[edit] Formal adjoint in one variable

[edit] Several variables

[edit] Example

[edit] Properties of differential operators

[edit] Several variables

[edit] Coordinate-independent description and relation to commutative algebra

[edit] Examples

[edit] See also

[edit] References