Editing Differential operator

{{Short description|Typically linear operator defined in terms of differentiation of functions}}
{{Use American English|date = January 2019}}
[[Image:Laplace's equation on an annulus.svg|right|thumb|300px|A harmonic function defined on an [[Annulus (mathematics)|annulus]]. Harmonic functions are exactly those functions which lie in the [[kernel (linear algebra)|kernel]] of the [[Laplace operator]], an important differential operator.]]

In [[mathematics]], a '''differential operator''' is an [[Operator (mathematics)|operator]] defined as a function of the [[derivative|differentiation]] operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a [[function (mathematics)|function]] and returns another function (in the style of a [[higher-order function]] in [[computer science]]).

This article considers mainly [[linear map|linear]] differential operators, which are the most common type. However, non-linear differential operators also exist, such as the [[Schwarzian derivative]].

==Definition==
Given a nonnegative integer ''m'', an order-<math>m</math> linear differential operator is a map <math>P</math> from a [[function space]] <math>\mathcal{F}_1</math> on <math>\mathbb{R}^n</math> to another function space <math>\mathcal{F}_2</math> that can be written as:

<math display="block">P = \sum_{|\alpha|\le m}a_\alpha(x) D^\alpha\ ,</math> 
where <math>\alpha = (\alpha_1,\alpha_2,\cdots,\alpha_n)</math> is a [[multi-index]] of non-negative [[integer]]s, <math>|\alpha| = \alpha_1 + \alpha_2 + \cdots + \alpha_n</math>, and for each <math>\alpha</math>, <math>a_\alpha(x)</math> is a function on some open domain in ''n''-dimensional space. The operator <math>D^\alpha</math> is interpreted as 

<math display="block">D^\alpha = \frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1}\partial x_2^{\alpha_2}\cdots\partial x_n^{\alpha_n}}</math> 

Thus for a function <math>f \in \mathcal{F}_1</math>:

<math display="block">P f = \sum_{|\alpha|\le m}a_\alpha(x) \frac{\partial^{|\alpha|}f}{\partial x_1^{\alpha_1}\partial x_2^{\alpha_2}\cdots\partial x_n^{\alpha_n}}</math>

The notation <math>D^{\alpha}</math> is justified (i.e., independent of order of differentiation) because of the [[symmetry of second derivatives]].

The polynomial ''p'' obtained by replacing partials <math>\frac{\partial}{\partial x_i}</math> by variables <math>\xi_i</math> in ''P'' is called the '''total symbol''' of ''P''; i.e., the total symbol of ''P'' above is:
<math display="block">p(x, \xi) = \sum_{|\alpha|\le m}a_\alpha(x) \xi^\alpha</math>
where <math>\xi^\alpha = \xi_1^{\alpha_1} \cdots \xi_n^{\alpha_n}.</math> The highest homogeneous component of the symbol, namely, 
:<math>\sigma(x, \xi) = \sum_{|\alpha|= m}a_\alpha(x) \xi^\alpha</math>
is called the '''principal symbol''' of ''P''.{{sfn|Hörmander|1983|p=151}} While the total symbol is not intrinsically defined, the principal symbol is intrinsically defined (i.e., it is a function on the cotangent bundle).<ref>{{harvnb|Schapira|1985|loc=1.1.7}}</ref>

More generally, let ''E'' and ''F'' be [[vector bundle]]s over a manifold ''X''. Then the linear operator

:<math> P: C^\infty(E) \to C^\infty(F) </math>

is a differential operator of order <math> k </math> if, in [[local coordinates]] on ''X'', we have 

:<math> Pu(x)  = \sum_{|\alpha| = k} P^\alpha(x) \frac {\partial^\alpha u} {\partial x^{\alpha}} + \text{lower-order terms}</math>

where, for each [[multi-index]] α, <math> P^\alpha(x):E \to F</math> is a [[bundle map]], symmetric on the indices α.

The ''k''<sup>th</sup> order coefficients of ''P'' transform as a [[symmetric tensor]]

:<math> \sigma_P: S^k (T^*X) \otimes E \to F </math>

whose domain is the [[tensor product]] of the ''k''<sup>th</sup> [[symmetric power]] of the [[cotangent bundle]] of ''X'' with ''E'', and whose codomain is ''F''.  This symmetric tensor is known as the '''principal symbol''' (or just the '''symbol''') of ''P''.

The coordinate system ''x''<sup>''i''</sup> permits a local trivialization of the cotangent bundle by the coordinate differentials d''x''<sup>''i''</sup>, which determine fiber coordinates ξ<sub>''i''</sub>. In terms of a basis of frames ''e''<sub>μ</sub>, ''f''<sub>ν</sub> of ''E'' and ''F'', respectively, the differential operator ''P'' decomposes into components

:<math>(Pu)_\nu = \sum_\mu P_{\nu\mu}u_\mu</math>

on each section ''u'' of ''E''.  Here ''P''<sub>νμ</sub> is the scalar differential operator defined by

:<math>P_{\nu\mu} = \sum_{\alpha} P_{\nu\mu}^\alpha\frac{\partial}{\partial x^\alpha}.</math>

With this trivialization, the principal symbol can now be written

:<math>(\sigma_P(\xi)u)_\nu = \sum_{|\alpha|=k} \sum_{\mu}P_{\nu\mu}^\alpha(x)\xi_\alpha u_\mu.</math>

In the cotangent space over a fixed point ''x'' of ''X'', the symbol <math> \sigma_P </math> defines a [[homogeneous polynomial]] of degree ''k'' in <math> T^*_x X </math> with values in <math> \operatorname{Hom}(E_x, F_x) </math>.

== Fourier interpretation ==
A differential operator ''P'' and its symbol appear naturally in connection with the [[Fourier transform]] as follows.  Let ƒ be a [[Schwartz function]].  Then by the inverse Fourier transform,

:<math>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.</math>

This exhibits ''P'' as a [[Fourier multiplier]].  A more general class of functions ''p''(''x'',ξ) which satisfy at most polynomial growth conditions in ξ under which this integral is well-behaved comprises the [[pseudo-differential operator]]s.

==Examples==
*The differential operator <math> P </math> is [[elliptic differential operator|elliptic]] if its symbol is invertible; that is for each nonzero <math> \theta \in T^*X </math> the bundle map <math> \sigma_P (\theta, \dots, \theta)</math> is invertible. On a [[compact manifold]], it follows from the elliptic theory that ''P'' is a [[Fredholm operator]]: it has finite-dimensional [[kernel (algebra)|kernel]] and cokernel.
*In the study of [[hyperbolic partial differential equation|hyperbolic]] and [[parabolic partial differential equation]]s, zeros of the principal symbol correspond to the [[method of characteristics|characteristics]] of the partial differential equation.
* In applications to the physical sciences, operators such as the [[Laplace operator]] play a major role in setting up and solving [[partial differential equation]]s.
* In [[differential topology]], the [[exterior derivative]] and [[Lie derivative]] operators have intrinsic meaning.
* In [[abstract algebra]], the concept of a [[derivation (abstract algebra)|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)]].
* In the development of [[holomorphic function]]s of a [[complex variable]] ''z'' = ''x'' + ''i''&thinsp;''y'', sometimes a complex function is considered to be a function of two real variables ''x'' and ''y''. Use is made of the [[Wirtinger derivative]]s, which are partial differential operators: <math display="block"> \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) \ .</math> This approach is also used to study functions of [[several complex variables]] and functions of a [[motor variable]].
*The differential operator [[del]], also called ''nabla'', is an important [[Euclidean vector|vector]] differential operator. It appears frequently in [[physics]] in places like the differential form of [[Maxwell's equations]]. In three-dimensional [[Cartesian coordinates]], del is defined as
:<math display="block">\nabla = \mathbf{\hat{x}} {\partial \over \partial x}  + \mathbf{\hat{y}} {\partial \over \partial y} + \mathbf{\hat{z}} {\partial \over \partial z}.</math>
:Del defines the [[gradient]], and is used to calculate the [[Curl (mathematics)|curl]], [[divergence]], and [[Laplacian]] of various objects.
*A [[chiral differential operator]]. For now, see [https://ncatlab.org/nlab/show/chiral+differential+operator]

==History==
The conceptual step of writing a differential operator as something free-standing is attributed to [[Louis François Antoine Arbogast]] in 1800.<ref>James Gasser (editor), ''A Boole Anthology: Recent and classical studies in the logic of George Boole'' (2000), p. 169; [https://books.google.com/books?id=A2Q5Yghl000C&pg=PA169 Google Books].</ref>

==Notations==
The most common differential operator is the action of taking the [[derivative]]. [[Notation for differentiation|Common notations]] for taking the first derivative with respect to a variable ''x'' include:

: <math>{d \over dx}</math>, <math>D</math>, <math>D_x,</math> and <math>\partial_x</math>.

When taking higher, ''n''th order derivatives, the operator may be written:

: <math>{d^n \over dx^n}</math>, <math>D^n</math>, <math>D^n_x</math>, or <math>\partial_x^n</math>.

The derivative of a function ''f'' of an [[argument of a function|argument]] ''x'' is sometimes given as either of the following:

: <math>[f(x)]'</math>
: <math>f'(x).</math>

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 equation]]s.

One of the most frequently seen differential operators is the [[Laplace operator|Laplacian operator]], defined by

:<math>\Delta = \nabla^2 = \sum_{k=1}^n \frac{\partial^2}{\partial x_k^2}.</math>

Another differential operator is the Θ operator, or [[theta operator]], defined by<ref>{{cite web| url=http://mathworld.wolfram.com/ThetaOperator.html|title=Theta Operator| author=E. W. Weisstein|access-date=2009-06-12}}</ref>

:<math>\Theta = z {d \over dz}.</math>

This is sometimes also called the '''homogeneity operator''', because its [[eigenfunction]]s are the [[monomial]]s in ''z'':
<math display="block">\Theta (z^k) = k z^k,\quad k=0,1,2,\dots </math>

In ''n'' variables the homogeneity operator is given by
<math display="block">\Theta = \sum_{k=1}^n x_k \frac{\partial}{\partial x_k}.</math>

As in one variable, the [[eigenspace]]s of Θ are the spaces of [[homogeneous function]]s. ([[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:
:<math>f \overleftarrow{\partial_x} g = g \cdot \partial_x f</math>
:<math>f \overrightarrow{\partial_x} g = f \cdot \partial_x g</math>
:<math>f \overleftrightarrow{\partial_x} g = f \cdot \partial_x g - g \cdot \partial_x f.</math>
Such a bidirectional-arrow notation is frequently used for describing the [[probability current]] of quantum mechanics.

==Adjoint of an operator==
{{See also|Hermitian adjoint}}
Given a linear differential operator <math>T</math>
<math display="block">Tu = \sum_{k=0}^n a_k(x) D^k u</math>
the [[Hermitian adjoint|adjoint]] of this operator is defined as the operator <math>T^*</math> such that
<math display="block">\langle Tu,v \rangle = \langle u, T^*v \rangle</math>
where the notation <math>\langle\cdot,\cdot\rangle</math> is used for the [[scalar product]] or [[inner product]].  This definition therefore depends on the definition of the scalar product (or inner product).

=== Formal adjoint in one variable ===

In the functional space of [[square-integrable function]]s on a [[real number|real]] [[interval (mathematics)|interval]] {{open-open|''a'', ''b''}}, the scalar product is defined by
<math display="block">\langle f, g \rangle = \int_a^b  \overline{f(x)} \,g(x) \,dx , </math>

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 <math>x \to a</math> and <math>x \to b</math>, one can also define the adjoint of ''T'' by
<math display="block">T^*u = \sum_{k=0}^n (-1)^k D^k \left[ \overline{a_k(x)} u \right].</math>

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 <math>T^*</math> is defined according to this formula, it is called the '''formal adjoint''' of ''T''.

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

=== Several variables ===

If Ω is a domain in '''R'''<sup>''n''</sup>, and ''P'' a differential operator on Ω, then the adjoint of ''P'' is defined in [[Lp space|''L''<sup>2</sup>(Ω)]] by duality in the analogous manner:

:<math>\langle f, P^* g\rangle_{L^2(\Omega)} = \langle P f, g\rangle_{L^2(\Omega)}</math>

for all smooth ''L''<sup>2</sup> functions ''f'', ''g''.  Since smooth functions are dense in ''L''<sup>2</sup>, this defines the adjoint on a dense subset of ''L''<sup>2</sup>:  P<sup>*</sup> is a [[densely defined operator]].

=== Example ===
The [[Sturm&ndash;Liouville theory|Sturm&ndash;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

: <math>Lu = -(pu')'+qu=-(pu''+p'u')+qu=-pu''-p'u'+qu=(-p) D^2 u +(-p') D u + (q)u.</math>

This property can be proven using the formal adjoint definition above.<ref>
: <math>\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}</math></ref>

This operator is central to [[Sturm–Liouville theory]] where the [[eigenfunctions]] (analogues to [[eigenvectors]]) of this operator are considered.

==Properties==

Differentiation is [[linear map|linear]], i.e.

:<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 [[function composition|compose]] differential operators by the rule

:<math>(D_1 \circ D_2)(f) = D_1(D_2(f)).</math>

Some care is then required: firstly any function coefficients in the operator ''D''<sub>2</sub> must be [[Differentiable function|differentiable]] as many times as the application of ''D''<sub>1</sub> requires. To get a [[ring (mathematics)|ring]] of such operators we must assume derivatives of all orders of the coefficients used. Secondly, this ring will not be [[commutative ring|commutative]]: an operator ''gD'' isn't the same in general as ''Dg''. For example we have the relation basic in [[quantum mechanics]]:
:<math>Dx - xD = 1.</math>

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]].

==Ring of polynomial differential operators==

===Ring of univariate polynomial differential operators===
{{Main|Weyl algebra}}

If ''R'' is a ring, let <math>R\langle D,X \rangle</math> be the [[non-commutative polynomial ring]] over ''R'' in the variables ''D'' and ''X'', and ''I'' the two-sided [[ideal (ring theory)|ideal]] generated by ''DX'' − ''XD'' − 1. Then the ring of univariate polynomial differential operators over ''R'' is the [[quotient ring]] <math>R\langle D,X\rangle/I</math>. This is a {{nowrap|non-commutative}} [[simple ring]]. Every element can be written in a unique way as a ''R''-linear combination of monomials of the form <math>X^a D^b \text{ mod } I</math>. It supports an analogue of [[Euclidean division of polynomials]].

Differential modules{{clarify|date=May 2021}} over <math>R[X]</math> (for the standard derivation) can be identified with [[module (mathematics)|modules]] over <math>R\langle D,X\rangle/I</math>.

===Ring of multivariate polynomial differential operators===

If ''R'' is a ring, let <math>R\langle D_1,\ldots,D_n,X_1,\ldots,X_n\rangle</math> be the non-commutative polynomial ring over ''R'' in the variables <math>D_1,\ldots,D_n,X_1,\ldots,X_n</math>, and ''I'' the two-sided ideal generated by the elements 
:<math>(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</math>
for all <math>1 \le i,j \le n,</math> where <math>\delta</math> is [[Kronecker delta]]. Then the ring of multivariate polynomial differential operators over ''R'' is the quotient ring {{nowrap|<math>R\langle D_1,\ldots,D_n,X_1,\ldots,X_n\rangle/I</math>.}}

This is a {{nowrap|non-commutative}} [[simple ring]].
Every element can be written in a unique way as a ''R''-linear combination of monomials of the form {{nowrap|<math>X_1^{a_1} \ldots X_n^{a_n} D_1^{b_1} \ldots D_n^{b_n}</math>.}}

==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 bundle]]s.  Let ''E'' and ''F'' be two vector bundles over a [[differentiable manifold]] ''M''. An '''R'''-linear mapping of [[vector bundle|sections]] {{nowrap|''P'' : Γ(''E'') → Γ(''F'')}} is said to be a '''''k''th-order linear differential operator''' if it factors through the [[jet bundle]] ''J''<sup>''k''</sup>(''E'').
In other words, there exists a linear mapping of vector bundles

:<math>i_P: J^k(E) \to F</math>

such that

:<math>P = i_P\circ j^k</math>

where {{nowrap|''j''<sup>''k''</sup>: Γ(''E'') → Γ(''J''<sup>''k''</sup>(''E''))}} is the prolongation that associates to any section of ''E'' its [[jet (mathematics)|''k''-jet]].

This just means that for a given [[vector bundle|section]] ''s'' of ''E'', the value of ''P''(''s'') at a point ''x''&nbsp;∈&nbsp;''M'' is fully determined by the ''k''th-order infinitesimal behavior of ''s'' in ''x''. In particular this implies that ''P''(''s'')(''x'') is determined by the [[sheaf (mathematics)|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===
{{Main Article|Differential calculus over commutative algebras}}

An equivalent, but purely algebraic description of linear differential operators is as follows: an '''R'''-linear map ''P'' is a ''k''th-order linear differential operator, if for any ''k''&nbsp;+&thinsp;1 smooth functions <math>f_0,\ldots,f_k \in C^\infty(M)</math> we have

:<math>[f_k,[f_{k-1},[\cdots[f_0,P]\cdots]]=0.</math>

Here the bracket <math>[f,P]:\Gamma(E)\to \Gamma(F)</math> is defined as the commutator

:<math>[f,P](s)=P(f\cdot s)-f\cdot P(s).</math>

This characterization of linear differential operators shows that they are particular mappings between [[module (mathematics)|modules]] over a [[commutative algebra (structure)|commutative algebra]], allowing the concept to be seen as a part of [[commutative algebra]].

== Variants ==
===A differential operator of infinite order ===
A differential operator of infinite order is (roughly) a differential operator whose total symbol is a [[power series]] instead of a polynomial.

=== Bidifferential operator ===
A differential operator acting on two functions <math>D(g,f)</math> is called a '''bidifferential operator'''. The notion appears, for instance, in an associative algebra structure on a deformation quantization of a Poisson algebra.<ref>{{cite journal |last1=Omori |first1=Hideki |last2=Maeda |first2=Y. |last3=Yoshioka |first3=A. |title=Deformation quantization of Poisson algebras |journal=Proceedings of the Japan Academy, Series A, Mathematical Sciences |date=1992 |volume=68 |issue=5 |doi=10.3792/PJAA.68.97 |s2cid=119540529 |language=en|doi-access=free }}</ref>

=== Microdifferential operator ===
A [[microdifferential 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.<ref>{{harvnb|Schapira|1985|loc=§ 1.2. § 1.3.}}</ref>

==See also==
{{div col|colwidth=22em}}
* [[Difference operator]]
* [[Delta operator]]
* [[Elliptic operator]]
* [[Curl (mathematics)]]
* [[Fractional calculus]]
* [[Invariant differential operator]]
* [[Differential calculus over commutative algebras]]
* [[Lagrangian system]]
* [[Spectral theory]]
* [[Energy operator]] 
* [[Momentum operator]]
* [[Pseudo-differential operator]]
* [[Fundamental solution]]
* [[Atiyah–Singer index theorem]] (section on symbol of operator)
* [[Malgrange–Ehrenpreis theorem]]
* [[Hypoelliptic operator]]
{{Div col end}}

== Notes == 
{{Reflist}}

== References ==
* {{citation|first=Daniel S.|last=Freed|title=Geometry of Dirac operators|page=8 |date=1987 |citeseerx=10.1.1.186.8445 }} 
* {{citation|mr=0717035|first=L.|last= Hörmander|authorlink=Lars Hörmander|title=The analysis of linear partial differential operators I|series= Grundl. Math. Wissenschaft. |volume= 256 |publisher=Springer |year=1983|isbn=3-540-12104-8 |doi=10.1007/978-3-642-96750-4}}.
*{{cite book |last1=Schapira |first1=Pierre |title=Microdifferential Systems in the Complex Domain |series=Grundlehren der mathematischen Wissenschaften |date=1985 |volume=269 |publisher=Springer |doi=10.1007/978-3-642-61665-5 |isbn=978-3-642-64904-2 |url=https://link.springer.com/book/10.1007/978-3-642-61665-5}}
* {{citation|last=Wells|first=R.O.|authorlink=Raymond O. Wells, Jr.|title=Differential analysis on complex manifolds|year=1973|publisher=Springer-Verlag|isbn=0-387-90419-0}}.

== Further reading ==
*{{cite journal |last1=Fedosov |first1=Boris |last2=Schulze |first2=Bert-Wolfgang |last3=Tarkhanov |first3=Nikolai |title=Analytic index formulas for elliptic corner operators |journal=Annales de l'Institut Fourier |date=2002 |volume=52 |issue=3 |pages=899–982 |doi=10.5802/aif.1906 |language=en |issn=1777-5310|doi-access=free }}
* https://mathoverflow.net/questions/451110/reference-request-inverse-of-differential-operators

==External links==
*{{Commons category-inline|Differential operators}}
*{{springer|title=Differential operator|id=p/d032250}}

{{Differential equations topics}}
{{Functional analysis}}
{{Authority control}}

[[Category:Operator theory]]
[[Category:Multivariable calculus]]
[[Category:Differential operators| ]]