Differential operator#Definition

Given a nonnegative integer m, an order-$m$ linear differential operator is a map $P$ from a function space $\backslash mathcal\{F\}\_1$ on $\backslash mathbb\{R\}^n$ to another function space $\backslash mathcal\{F\}\_2$ that can be written as:

$$P\; =\; \backslash sum\_$$

Thus for a function $f\; \backslash in\; \backslash mathcal\{F\}\_1$:

$$P\; f\; =\; \backslash sum\_$$

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

The polynomial p obtained by replacing partials $\backslash frac\{\backslash partial\}\{\backslash partial\; x\_i\}$ by variables $\backslash xi\_i$ in P is called the total symbol of P; i.e., the total symbol of P above is:

$$p(x,\; \backslash xi)\; =\; \backslash sum\_\{|\backslash alpha|\backslash le\; m\}a\_\backslash alpha(x)\; \backslash xi^\backslash alpha$$

where $\backslash xi^\backslash alpha\; =\; \backslash xi\_1^\{\backslash alpha\_1\}\; \backslash cdots\; \backslash xi\_n^\{\backslash alpha\_n\}.$ The highest homogeneous component of the symbol, namely,

:$\backslash sigma(x,\; \backslash xi)\; =\; \backslash sum\_\{|\backslash alpha|=\; m\}a\_\backslash alpha(x)\; \backslash xi^\backslash alpha$

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).{{harvnb|Schapira|1985|loc=1.1.7}}

More generally, let E and F be vector bundles over a manifold X. Then the linear operator

:$P:\; C^\backslash infty(E)\; \backslash to\; C^\backslash infty(F)$

is a differential operator of order $k$ if, in local coordinates on X, we have

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

where, for each multi-index α, $P^\backslash alpha(x):E\; \backslash to\; F$ is a bundle map, symmetric on the indices α.

The k^th order coefficients of P transform as a symmetric tensor

:$\backslash sigma\_P:\; S^k\; (T^*X)\; \backslash otimes\; E\; \backslash to\; F$

whose domain is the tensor product of the k^th 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 xⁱ 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 frames e_μ, f_ν of E and F, respectively, the differential operator P decomposes into components

:$(Pu)\_\backslash nu\; =\; \backslash sum\_\backslash mu\; P\_\{\backslash nu\backslash mu\}u\_\backslash mu$

on each section u of E. Here P_νμ is the scalar differential operator defined by

:$P\_\{\backslash nu\backslash mu\}\; =\; \backslash sum\_\{\backslash alpha\}\; P\_\{\backslash nu\backslash mu\}^\backslash alpha\backslash frac\{\backslash partial\}\{\backslash partial\; x^\backslash alpha\}.$

With this trivialization, the principal symbol can now be written

:$(\backslash sigma\_P(\backslash xi)u)\_\backslash nu\; =\; \backslash sum\_\{|\backslash alpha|=k\}\; \backslash sum\_\{\backslash mu\}P\_\{\backslash nu\backslash mu\}^\backslash alpha(x)\backslash xi\_\backslash alpha\; u\_\backslash mu.$

In the cotangent space over a fixed point x of X, the symbol $\backslash sigma\_P$ defines a homogeneous polynomial of degree k in $T^*\_x\; X$ with values in $\backslash operatorname\{Hom\}(E\_x,\; F\_x)$.

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,

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

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 operators.

• The differential operator $P$ is elliptic if its symbol is invertible; that is for each nonzero $\backslash theta\; \backslash in\; T^*X$ the bundle map $\backslash sigma\_P\; (\backslash theta,\; \backslash dots,\; \backslash theta)$ is invertible. On a compact manifold, it follows from the elliptic theory that P is a Fredholm operator: it has finite-dimensional kernel and cokernel.
• In the study of hyperbolic and parabolic partial differential equations, zeros of the principal symbol correspond to the 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 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).
• In the development of holomorphic functions of a complex variable z = x + i y, sometimes a complex function is considered to be a function of two real variables x and y. Use is made of the Wirtinger derivatives, which are partial differential operators: $$\backslash frac\{\backslash partial\}\{\backslash partial\; z\}\; =\; \backslash frac\{1\}\{2\}\; \backslash left(\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\}\; -\; i\; \backslash frac\{\backslash partial\}\{\backslash partial\; y\}\; \backslash right)\; \backslash \; ,\backslash quad\; \backslash frac\{\backslash partial\}\{\backslash partial\backslash bar\{z\}\}=\; \backslash frac\{1\}\{2\}\; \backslash left(\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\}\; +\; i\; \backslash frac\{\backslash partial\}\{\backslash partial\; y\}\; \backslash right)\; \backslash \; .$$ 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 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

:$$\backslash nabla\; =\; \backslash mathbf\{\backslash hat\{x\}\}\; \{\backslash partial\; \backslash over\; \backslash partial\; x\}\; +\; \backslash mathbf\{\backslash hat\{y\}\}\; \{\backslash partial\; \backslash over\; \backslash partial\; y\}\; +\; \backslash mathbf\{\backslash hat\{z\}\}\; \{\backslash partial\; \backslash over\; \backslash partial\; z\}.$$

:Del defines the gradient, and is used to calculate the curl, divergence, and Laplacian of various objects.

• A chiral differential operator. For now, see [https://ncatlab.org/nlab/show/chiral+differential+operator]

The conceptual step of writing a differential operator as something free-standing is attributed to Louis François Antoine Arbogast in 1800.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].

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

: $\{d\; \backslash over\; dx\}$, $D$, $D\_x,$ and $\backslash partial\_x$.

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

: $\{d^n\; \backslash over\; dx^n\}$, $D^n$, $D^n\_x$, or $\backslash partial\_x^n$.

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

: $[f(x)]\text{'}$

: $f\text{'}(x).$

The D notation's use and creation is credited to Oliver Heaviside, who considered differential operators of the form

: $\backslash sum\_\{k=0\}^n\; c\_k\; D^k$

in his study of differential equations.

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

:$\backslash Delta\; =\; \backslash nabla^2\; =\; \backslash sum\_\{k=1\}^n\; \backslash frac\{\backslash partial^2\}\{\backslash partial\; x\_k^2\}.$

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

:$\backslash Theta\; =\; z\; \{d\; \backslash over\; dz\}.$

This is sometimes also called the homogeneity operator, because its eigenfunctions are the monomials in z:

$$\backslash Theta\; (z^k)\; =\; k\; z^k,\backslash quad\; k=0,1,2,\backslash dots$$

In n variables the homogeneity operator is given by

$$\backslash Theta\; =\; \backslash sum\_\{k=1\}^n\; x\_k\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\_k\}.$$

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

:$f\; \backslash overleftarrow\{\backslash partial\_x\}\; g\; =\; g\; \backslash cdot\; \backslash partial\_x\; f$

:$f\; \backslash overrightarrow\{\backslash partial\_x\}\; g\; =\; f\; \backslash cdot\; \backslash partial\_x\; g$

:$f\; \backslash overleftrightarrow\{\backslash partial\_x\}\; g\; =\; f\; \backslash cdot\; \backslash partial\_x\; g\; -\; g\; \backslash cdot\; \backslash partial\_x\; f.$

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

{{See also|Hermitian adjoint}}

Given a linear differential operator $T$

$$Tu\; =\; \backslash sum\_\{k=0\}^n\; a\_k(x)\; D^k\; u$$

the adjoint of this operator is defined as the operator $T^*$ such that

$$\backslash langle\; Tu,v\; \backslash rangle\; =\; \backslash langle\; u,\; T^*v\; \backslash rangle$$

where the notation $\backslash langle\backslash cdot,\backslash cdot\backslash rangle$ is used for the scalar product or inner product. This definition therefore depends on the definition of the scalar product (or inner product).

In the functional space of square-integrable functions on a real interval {{open-open|a, b}}, the scalar product is defined by

$$\backslash langle\; f,\; g\; \backslash rangle\; =\; \backslash int\_a^b\; \backslash overline\{f(x)\}\; \backslash ,g(x)\; \backslash ,dx\; ,$$

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 $x\; \backslash to\; a$ and $x\; \backslash to\; b$, one can also define the adjoint of T by

$$T^*u\; =\; \backslash sum\_\{k=0\}^n\; (-1)^k\; D^k\; \backslash left[\; \backslash overline\{a\_k(x)\}\; u\; \backslash right].$$

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 the formal adjoint of T.

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

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:

:$\backslash langle\; f,\; P^*\; g\backslash rangle\_\{L^2(\backslash Omega)\}\; =\; \backslash langle\; P\; f,\; g\backslash rangle\_\{L^2(\backslash Omega)\}$

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.

The Sturm–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

: $Lu\; =\; -(pu\text{'})\text{'}+qu=-(pu$+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.

: $\backslash begin\{align\}$

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 \\

& {} = -pu-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.

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

:$(D\_1\; \backslash circ\; D\_2)(f)\; =\; D\_1(D\_2(f)).$

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. For example we have 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.

{{Main|Weyl algebra}}

If R is a ring, let $R\backslash langle\; D,X\; \backslash rangle$ be the non-commutative polynomial ring over R in the variables D and X, and I the two-sided ideal generated by DX − XD − 1. Then the ring of univariate polynomial differential operators over R is the quotient ring $R\backslash langle\; D,X\backslash rangle/I$. 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 $X^a\; D^b\; \backslash text\{\; mod\; \}\; I$. It supports an analogue of Euclidean division of polynomials.

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

If R is a ring, let $R\backslash langle\; D\_1,\backslash ldots,D\_n,X\_1,\backslash ldots,X\_n\backslash rangle$ be the non-commutative polynomial ring over R in the variables $D\_1,\backslash ldots,D\_n,X\_1,\backslash ldots,X\_n$, and I the two-sided ideal generated by the elements

:$(D\_i\; X\_j-X\_j\; D\_i)-\backslash delta\_\{i,j\},\backslash \; \backslash \; \backslash \; D\_i\; D\_j\; -D\_j\; D\_i,\backslash \; \backslash \; \backslash \; X\_i\; X\_j\; -\; X\_j\; X\_i$

for all $1\; \backslash le\; i,j\; \backslash le\; n,$ where $\backslash delta$ is Kronecker delta. Then the ring of multivariate polynomial differential operators over R is the quotient ring {{nowrap|$R\backslash langle\; D\_1,\backslash ldots,D\_n,X\_1,\backslash ldots,X\_n\backslash rangle/I$.}}

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|$X\_1^\{a\_1\}\; \backslash ldots\; X\_n^\{a\_n\}\; D\_1^\{b\_1\}\; \backslash ldots\; D\_n^\{b\_n\}$.}}

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 {{nowrap|P : Γ(E) → Γ(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

:$i\_P:\; J^k(E)\; \backslash to\; F$

such that

:$P\; =\; i\_P\backslash circ\; j^k$

where {{nowrap|j^k: Γ(E) → Γ(J^k(E))}} is the prolongation that associates to any section of E its k-jet.

This just means that for a given section 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.

{{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 kth-order linear differential operator, if for any k + 1 smooth functions $f\_0,\backslash ldots,f\_k\; \backslash in\; C^\backslash infty(M)$ we have

:$[f\_k,[f\_\{k-1\},[\backslash cdots[f\_0,P]\backslash cdots]]=0.$

Here the bracket $[f,P]:\backslash Gamma(E)\backslash to\; \backslash Gamma(F)$ is defined as the commutator

:$[f,P](s)=P(f\backslash cdot\; s)-f\backslash cdot\; P(s).$

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.

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

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

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.{{harvnb|Schapira|1985|loc=§ 1.2. § 1.3.}}

{{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}}

• {{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}}.

• {{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

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

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Category:Operator theory

Category:Multivariable calculus