Differential operator
|
|
In mathematics, a differential operator is a linear 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).
| Contents |
Notations
The most commonly used differential operator is the action of taking the derivative itself. Common notations for this operator include:
- <math>{d \over dx}<math>
- <math>D<math>, where the variable one is differentiating to is clear, and
- <math>D_t<math>, where the variable is declared explicitly.
First derivatives are signified as above, but when taking higher, n-th derivatives, the following alterations are useful:
- <math>d^n \over dx^n<math>
- <math>D^n<math>
- <math>D^n_t<math>
Operator methods
The D notation's use and creation is credited to Oliver Heaviside, who considered differential operators of the form
- <math>\sum_{k=0}^n c_k D^k<math>
in his study of differential equations.
Given a linear differential operator
- <math>Tu = \sum_{k=0}^n a_k(x) D^k u<math>
the adjoint of this operator is defined as
- <math>T^*u = \sum_{k=0}^n (-1)^k D^k (a_k(x)u)<math>
Second order linear formally self-adjoint differential operators L can be written in the form
- <math>Lu = -(pu')'+qu=-(pu''+p'u')+qu=-pu''-p'u'+qu=(-p) D^2 u +(-p') D u + (q)u\;\!<math>
since using the adjoint definition above,
- <math>\begin{matrix}
L^*u &=& (-1)^2 D^2 ((-p)u) + (-1)^1 D ((-p')u) + (-1)^0 (qu) \\
&=& D^2(-pu) - D(-p'u) + qu \\ &=& -D^2(pu) + D(p'u)+qu \\ &=& -(pu)''+(p'u)'+qu \\ &=& -(p'u+pu')'+(p''u+p'u')+qu \\ &=& -(p'u)'-(pu')'+p''u+p'u'+qu \\ &=& -(p''u+p'u')-(p'u'+pu'')+p''u+p'u'+qu \\ &=& -p''u-p'u'-p'u'-pu''+p''u+p'u'+qu \\ &=& -p''u+p''u-p'u'-p'u'+p'u'-pu''+qu \\ &=& -p'u'-pu''+qu \\ &=& -(pu')'+qu\quad\square \\
\end{matrix}<math>
This operator is central to Sturm-Liouville theory where the eigenfunctions (analogues to eigenvectors) of this operator are considered.
This operator L is known as formally self-adjoint, different from the usage to self-adjoint operators in Hilbert spaces, in that if we define the inner product
- <math>\langle f, g \rangle = \int_a^b fg \,dx<math>
- <math>\langle u,Lv \rangle = \langle v, Lu \rangle<math>
- <math>\int_a^b uLv \,dx = \int_a^b vLu \,dx<math>
we desire the evaluation at the endpoints to be zero at infinity, which only occurs under certain conditions.
Other forms
One of the most frequently seen differential operators is the Laplacian operator
- <math>\Delta=\nabla^{2}=\sum_{k=1}^n {\partial^2\over \partial x_k^2}.<math>
Properties of differential operators
Differentiation is linear, ie
- <math>D (f+g) = (Df) + (Dg)<math>
- <math>D (af) = a (Df)<math>
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
- (D1oD2)(f) = D1 (D2(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.
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 manifold M. An operator is a mapping of sections, <math>P:\Gamma(E)\rightarrow\Gamma(F)<math> which maps the stalk of the sheaf of germs of <math>\Gamma(E)<math> at a point <math>x\in M<math> to the fibre of F at x:
- <math>P:\Gamma_x(E)\rightarrow F_x<math>.
An operator P is said to be a k-th order differential operator if it factors through the jet bundle <math>J^k(E)<math>. In other words, there exists a linear mapping of vector bundles
- <math>i_P:J^k(E)\rightarrow F<math>
such that <math>P=i_P\circ j^k<math> as in the following composition:
- <math>P:\Gamma_x(E)\rightarrow J^k(E)_x\rightarrow F_x<math>.
Examples
In applications to the physical sciences, operators such as the Laplace operator play a major role via the setting up and solution of partial differential equations.
In differential geometry the exterior derivative and Lie derivative operators have intrinsic meaning.
In abstract algebra the concept of derivation means that differential operators may still be defined, in the absence of calculus concepts based on geometry.