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Differential operator

From Wikipedia, the free encyclopedia

Typically linear operator defined in terms of differentiation of functions

A harmonic function defined on anannulus. Harmonic functions are exactly those functions which lie in thekernel of theLaplace operator, an important differential operator.

Inmathematics, adifferential operator is anoperator defined as a function of thedifferentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts afunction and returns another function (in the style of ahigher-order function incomputer science).

This article considers mainlylinear differential operators, which are the most common type. However, non-linear differential operators also exist, such as theSchwarzian derivative.

Definition

Given a nonnegative integerm, an order- $m {\displaystyle m}$ linear differential operator is a map $P {\displaystyle P}$ from afunction space ${\mathcal {F}}_{1}$ on $\mathbb {R} ^{n}$ to another function space ${\mathcal {F}}_{2}$ that can be written as:

$P=\sum _{|\alpha |\leq m}a_{\alpha }(x)D^{\alpha }\ ,$ where $\alpha =(\alpha _{1},\alpha _{2},\cdots ,\alpha _{n})$ is amulti-index of non-negativeintegers, $|\alpha |=\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}$ , and for each $\alpha$ , $a_{\alpha }(x)$ is a function on some open domain inn-dimensional space. The operator $D^{\alpha }$ is interpreted as

$D^{\alpha }={\frac {\partial ^{|\alpha |}}{\partial x_{1}^{\alpha _{1}}\partial x_{2}^{\alpha _{2}}\cdots \partial x_{n}^{\alpha _{n}}}}$

Thus for a function $f\in {\mathcal {F}}_{1}$ :

$Pf=\sum _{|\alpha |\leq m}a_{\alpha }(x){\frac {\partial ^{|\alpha |}f}{\partial x_{1}^{\alpha _{1}}\partial x_{2}^{\alpha _{2}}\cdots \partial x_{n}^{\alpha _{n}}}}$

The notation $D^{\alpha }$ is justified (i.e., independent of order of differentiation) because of thesymmetry of second derivatives.

The polynomialp obtained by replacing partials ${\frac {\partial }{\partial x_{i}}}$ by variables $\xi _{i}$ inP is called thetotal symbol ofP; i.e., the total symbol ofP above is: $p(x,\xi )=\sum _{|\alpha |\leq m}a_{\alpha }(x)\xi ^{\alpha }$ where $\xi ^{\alpha }=\xi _{1}^{\alpha _{1}}\cdots \xi _{n}^{\alpha _{n}}.$ The highest homogeneous component of the symbol, namely,

\sigma (x,\xi )=\sum _{|\alpha |=m}a_{\alpha }(x)\xi ^{\alpha }

is called theprincipal symbol ofP.^[1] While the total symbol is not intrinsically defined, the principal symbol is intrinsically defined (i.e., it is a function on the cotangent bundle).^[2]

More generally, letE andF bevector bundles over a manifoldX. Then the linear operator

P:C^{\infty }(E)\to C^{\infty }(F)

is a differential operator of order $k {\displaystyle k}$ if, inlocal coordinates onX, we have

Pu(x)=\sum _{|\alpha |=k}P^{\alpha }(x){\frac {\partial ^{\alpha }u}{\partial x^{\alpha }}}+{\text{lower-order terms}}

where, for eachmulti-index α, $P^{\alpha }(x):E\to F$ is abundle map, symmetric on the indices α.

Thek^th order coefficients ofP transform as asymmetric tensor

\sigma _{P}:S^{k}(T^{*}X)\otimes E\to F

whose domain is thetensor product of thek^thsymmetric power of thecotangent bundle ofX withE, and whose codomain isF. This symmetric tensor is known as theprincipal symbol (or just thesymbol) ofP.

The coordinate systemxⁱ permits a local trivialization of the cotangent bundle by the coordinate differentials dxⁱ, which determine fiber coordinates ξ_i. In terms of a basis of framese_μ,f_ν ofE andF, respectively, the differential operatorP decomposes into components

(Pu)_{\nu }=\sum _{\mu }P_{\nu \mu }u_{\mu }

on each sectionu ofE. HereP_νμ is the scalar differential operator defined by

P_{\nu \mu }=\sum _{\alpha }P_{\nu \mu }^{\alpha }{\frac {\partial }{\partial x^{\alpha }}}.

With this trivialization, the principal symbol can now be written

(\sigma _{P}(\xi )u)_{\nu }=\sum _{|\alpha |=k}\sum _{\mu }P_{\nu \mu }^{\alpha }(x)\xi _{\alpha }u_{\mu }.

In the cotangent space over a fixed pointx ofX, the symbol $\sigma _{P}$ defines ahomogeneous polynomial of degreek in $T_{x}^{*}X$ with values in $\operatorname {Hom} (E_{x},F_{x})$ .

Fourier interpretation

A differential operatorP and its symbol appear naturally in connection with theFourier transform as follows. Let ƒ be aSchwartz function. Then by the inverse Fourier transform,

Pf(x)={\frac {1}{(2\pi )^{\frac {d}{2}}}}\int \limits _{\mathbf {R} ^{d}}e^{ix\cdot \xi }p(x,i\xi ){\hat {f}}(\xi )\,d\xi .

This exhibitsP as aFourier multiplier. A more general class of functionsp(x,ξ) which satisfy at most polynomial growth conditions in ξ under which this integral is well-behaved comprises thepseudo-differential operators.

Examples

The differential operator $P {\displaystyle P}$ iselliptic if its symbol is invertible; that is for each nonzero $\theta \in T^{*}X$ the bundle map $\sigma _{P}(\theta ,\dots ,\theta )$ is invertible. On acompact manifold, it follows from the elliptic theory thatP is aFredholm operator: it has finite-dimensionalkernel and cokernel.
In the study ofhyperbolic andparabolic partial differential equations, zeros of the principal symbol correspond to thecharacteristics of the partial differential equation.
In applications to the physical sciences, operators such as theLaplace operator play a major role in setting up and solvingpartial differential equations.
Indifferential topology, theexterior derivative andLie derivative operators have intrinsic meaning.
Inabstract algebra, the concept of aderivation allows for generalizations of differential operators, which do not require the use of calculus. Frequently such generalizations are employed inalgebraic geometry andcommutative algebra. See alsoJet (mathematics).
In the development ofholomorphic functions of acomplex variablez =x +i y, sometimes a complex function is considered to be a function of two real variablesx andy. Use is made of theWirtinger derivatives, which are partial differential operators: ${\frac {\partial }{\partial z}}={\frac {1}{2}}\left({\frac {\partial }{\partial x}}-i{\frac {\partial }{\partial y}}\right)\ ,\quad {\frac {\partial }{\partial {\bar {z}}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right)\ .$ This approach is also used to study functions ofseveral complex variables and functions of amotor variable.
The differential operatordel, also callednabla, is an importantvector differential operator. It appears frequently inphysics in places like the differential form ofMaxwell's equations. In three-dimensionalCartesian coordinates, del is defined as

\nabla =\mathbf {\hat {x}} {\partial  \over \partial x}+\mathbf {\hat {y}} {\partial  \over \partial y}+\mathbf {\hat {z}} {\partial  \over \partial z}.

Del defines thegradient, and is used to calculate thecurl,divergence, andLaplacian of various objects.

Achiral differential operator. For now, see[1]

History

The conceptual step of writing a differential operator as something free-standing is attributed toLouis François Antoine Arbogast in 1800.^[3]

Notations

The most common differential operator is the action of taking thederivative.Common notations for taking the first derivative with respect to a variablex include:

{d \over dx}

,

D {\displaystyle D}

,

D_{x},

and

\partial _{x}

.

When taking higher,nth order derivatives, the operator may be written:

{d^{n} \over dx^{n}}

,

D^{n}

,

D_{x}^{n}

, or

\partial _{x}^{n}

.

The derivative of a functionf of anargumentx is sometimes given as either of the following:

[f(x)]'

f'(x).

TheD notation's use and creation is credited toOliver Heaviside, who considered differential operators of the form

\sum _{k=0}^{n}c_{k}D^{k}

in his study ofdifferential equations.

One of the most frequently seen differential operators is theLaplacian operator, defined by

\Delta =\nabla ^{2}=\sum _{k=1}^{n}{\frac {\partial ^{2}}{\partial x_{k}^{2}}}.

Another differential operator is the Θ operator, ortheta operator, defined by^[4]

\Theta =z{d \over dz}.

This is sometimes also called thehomogeneity operator, because itseigenfunctions are themonomials inz: $\Theta (z^{k})=kz^{k},\quad k=0,1,2,\dots$

Inn variables the homogeneity operator is given by $\Theta =\sum _{k=1}^{n}x_{k}{\frac {\partial }{\partial x_{k}}}.$

As in one variable, theeigenspaces of Θ are the spaces ofhomogeneous functions. (Euler's homogeneous function theorem)

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:

f{\overleftarrow {\partial _{x}}}g=g\cdot \partial _{x}f

f{\overrightarrow {\partial _{x}}}g=f\cdot \partial _{x}g

f{\overleftrightarrow {\partial _{x}}}g=f\cdot \partial _{x}g-g\cdot \partial _{x}f.

Such a bidirectional-arrow notation is frequently used for describing theprobability current of quantum mechanics.

Adjoint of an operator

See also:Hermitian adjoint

Given a linear differential operator $T {\displaystyle T}$ $Tu=\sum _{k=0}^{n}a_{k}(x)D^{k}u$ theadjoint 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 thescalar product orinner product. This definition therefore depends on the definition of the scalar product (or inner product).

Formal adjoint in one variable

In the functional space ofsquare-integrable functions on areal interval(a,b), the scalar product is defined by $\langle f,g\rangle =\int _{a}^{b}{\overline {f(x)}}\,g(x)\,dx,$

where the line overf(x) denotes thecomplex conjugate off(x). If one moreover adds the condition thatf org vanishes as $x\to a$ and $x\to b$ , one can also define the adjoint ofT by $T^{*}u=\sum _{k=0}^{n}(-1)^{k}D^{k}\left[{\overline {a_{k}(x)}}u\right].$

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 theformal adjoint ofT.

A (formally)self-adjoint operator is an operator equal to its own (formal) adjoint.

Several variables

If Ω is a domain inRⁿ, andP a differential operator on Ω, then the adjoint ofP is defined inL²(Ω) by duality in the analogous manner:

\langle f,P^{*}g\rangle _{L^{2}(\Omega )}=\langle Pf,g\rangle _{L^{2}(\Omega )}

for all smoothL² functionsf,g. Since smooth functions are dense inL², this defines the adjoint on a dense subset ofL²: P^* is adensely defined operator.

Example

TheSturm–Liouville operator is a well-known example of a formal self-adjoint operator. This second-order linear differential operatorL can be written in the form

Lu=-(pu')'+qu=-(pu''+p'u')+qu=-pu''-p'u'+qu=(-p)D^{2}u+(-p')Du+(q)u.

This property can be proven using the formal adjoint definition above.^[5]

This operator is central toSturm–Liouville theory where theeigenfunctions (analogues toeigenvectors) of this operator are considered.

Properties

Differentiation islinear, i.e.

D(f+g)=(Df)+(Dg),

D(af)=a(Df),

wheref andg are functions, anda is a constant.

Anypolynomial inD with function coefficients is also a differential operator. We may alsocompose 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 operatorD₂ must bedifferentiable as many times as the application ofD₁ requires. To get aring of such operators we must assume derivatives of all orders of the coefficients used. Secondly, this ring will not becommutative: an operatorgD isn't the same in general asDg. For example we have the relation basic inquantum mechanics:

Dx-xD=1.

The subring of operators that are polynomials inD withconstant coefficients is, by contrast, commutative. It can be characterised another way: it consists of the translation-invariant operators.

The differential operators also obey theshift theorem.

Ring of polynomial differential operators

Ring of univariate polynomial differential operators

Main article:Weyl algebra

IfR is a ring, let $R\langle D,X\rangle$ be thenon-commutative polynomial ring overR in the variablesD andX, andI the two-sidedideal generated byDX −XD − 1. Then the ring of univariate polynomial differential operators overR is thequotient ring $R\langle D,X\rangle /I$ . This is anon-commutativesimple ring. Every element can be written in a unique way as aR-linear combination of monomials of the form $X^{a}D^{b}{\text{ mod }}I$ . It supports an analogue ofEuclidean division of polynomials.

Differential modules^{[clarification needed]} over $R[X]$ (for the standard derivation) can be identified withmodules over $R\langle D,X\rangle /I$ .

Ring of multivariate polynomial differential operators

IfR is a ring, let $R\langle D_{1},\ldots ,D_{n},X_{1},\ldots ,X_{n}\rangle$ be the non-commutative polynomial ring overR in the variables $D_{1},\ldots ,D_{n},X_{1},\ldots ,X_{n}$ , andI the two-sided ideal generated by the elements

(D_{i}X_{j}-X_{j}D_{i})-\delta _{i,j},\ \ \ D_{i}D_{j}-D_{j}D_{i},\ \ \ X_{i}X_{j}-X_{j}X_{i}

for all $1\leq i,j\leq n,$ where $\delta$ isKronecker delta. Then the ring of multivariate polynomial differential operators overR is the quotient ring $R\langle D_{1},\ldots ,D_{n},X_{1},\ldots ,X_{n}\rangle /I$ .

This is anon-commutativesimple ring.Every element can be written in a unique way as aR-linear combination of monomials of the form $X_{1}^{a_{1}}\ldots X_{n}^{a_{n}}D_{1}^{b_{1}}\ldots D_{n}^{b_{n}}$ .

Coordinate-independent description

Indifferential geometry andalgebraic geometry it is often convenient to have acoordinate-independent description of differential operators between twovector bundles. LetE andF be two vector bundles over adifferentiable manifoldM. AnR-linear mapping ofsectionsP : Γ(E) → Γ(F) is said to be akth-order linear differential operator if it factors through thejet bundleJ^k(E).In other words, there exists a linear mapping of vector bundles

i_{P}:J^{k}(E)\to F

such that

P=i_{P}\circ j^{k}

wherej^k: Γ(E) → Γ(J^k(E)) is the prolongation that associates to any section ofE itsk-jet.

This just means that for a givensections ofE, the value ofP(s) at a pointx ∈ M is fully determined by thekth-order infinitesimal behavior ofs inx. In particular this implies thatP(s)(x) is determined by thegerm ofs inx, which is expressed by saying that differential operators are local. A foundational result is thePeetre theorem showing that the converse is also true: any (linear) local operator is differential.

Relation to commutative algebra

Main article:Differential calculus over commutative algebras

An equivalent, but purely algebraic description of linear differential operators is as follows: anR-linear mapP is akth-order linear differential operator, if for anyk + 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 $[f,P]:\Gamma (E)\to \Gamma (F)$ 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 betweenmodules over acommutative algebra, allowing the concept to be seen as a part ofcommutative algebra.

Variants

A differential operator of infinite order

A differential operator of infinite order is (roughly) a differential operator whose total symbol is apower series instead of a polynomial.

Invariant differential operator

Aninvariant differential operator is a differential operator that is also an invariant operator (e.g., commutes with a group action).

Bidifferential operator

A differential operator acting on two functions $D(g,f)$ is called abidifferential operator. The notion appears, for instance, in an associative algebra structure on a deformation quantization of a Poisson algebra.^[6]

Microdifferential operator

Amicrodifferential operator is a type of operator on an open subset of a cotangent bundle, as opposed to an open subset of a manifold. It is obtained by extending the notion of a differential operator to the cotangent bundle.^[7]

See also

Notes

^Hörmander 1983, p. 151.
^Schapira 1985, 1.1.7
^James Gasser (editor),A Boole Anthology: Recent and classical studies in the logic of George Boole (2000), p. 169;Google Books.
^E. W. Weisstein."Theta Operator". Retrieved2009-06-12.
^
${\begin{aligned}L^{*}u&{}=(-1)^{2}D^{2}[(-p)u]+(-1)^{1}D[(-p')u]+(-1)^{0}(qu)\\&{}=-D^{2}(pu)+D(p'u)+qu\\&{}=-(pu)''+(p'u)'+qu\\&{}=-p''u-2p'u'-pu''+p''u+p'u'+qu\\&{}=-p'u'-pu''+qu\\&{}=-(pu')'+qu\\&{}=Lu\end{aligned}}$
^Omori, Hideki; Maeda, Y.; Yoshioka, A. (1992)."Deformation quantization of Poisson algebras".Proceedings of the Japan Academy, Series A, Mathematical Sciences.68 (5).doi:10.3792/PJAA.68.97.S2CID 119540529.
^Schapira 1985, § 1.2. § 1.3.

References

Freed, Daniel S. (1987),Geometry of Dirac operators, p. 8,CiteSeerX 10.1.1.186.8445
Hörmander, L. (1983),The analysis of linear partial differential operators I, Grundl. Math. Wissenschaft., vol. 256, Springer,doi:10.1007/978-3-642-96750-4,ISBN 3-540-12104-8,MR 0717035.
Schapira, Pierre (1985).Microdifferential Systems in the Complex Domain. Grundlehren der mathematischen Wissenschaften. Vol. 269. Springer.doi:10.1007/978-3-642-61665-5.ISBN 978-3-642-64904-2.
Wells, R.O. (1973),Differential analysis on complex manifolds, Springer-Verlag,ISBN 0-387-90419-0.

Further reading

Fedosov, Boris; Schulze, Bert-Wolfgang; Tarkhanov, Nikolai (2002)."Analytic index formulas for elliptic corner operators".Annales de l'Institut Fourier.52 (3):899–982.doi:10.5802/aif.1906.ISSN 1777-5310.
https://mathoverflow.net/questions/451110/reference-request-inverse-of-differential-operators

External links

Media related toDifferential operators at Wikimedia Commons
"Differential operator",Encyclopedia of Mathematics,EMS Press, 2001 [1994]

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Operations	Differential operator Notation for differentiation Ordinary Partial Differential-algebraic Integro-differential Fractional Linear Non-linear Holonomic
Attributes of variables	Dependent and independent variables Homogeneous Nonhomogeneous Coupled Decoupled Order Degree Autonomous Exact differential equation On jet bundles
Relation to processes	Difference (discrete analogue) Stochastic Stochastic partial Delay

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