Formal Adjoint
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In mathematics, a differential operator is an
operator Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another ...
defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includin ...
). This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative.


Definition

An order-m linear differential operator is a map A from a function space \mathcal_1 to another function space \mathcal_2 that can be written as: A = \sum_a_\alpha(x) D^\alpha\ , where \alpha = (\alpha_1,\alpha_2,\cdots,\alpha_n) is a multi-index of non-negative
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s, , \alpha, = \alpha_1 + \alpha_2 + \cdots + \alpha_n, and for each \alpha, a_\alpha(x) is a function on some open domain in ''n''-dimensional space. The operator D^\alpha is interpreted as D^\alpha = \frac Thus for a function f \in \mathcal_1: A f = \sum_a_\alpha(x) \frac A differential operator acting on two functions D(g,f) is also called a ''bidifferential operator''.


Notations

The most common differential operator is the action of taking the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
. Common notations for taking the first derivative with respect to a variable ''x'' include: : , D, D_x, and \partial_x. When taking higher, ''n''th order derivatives, the operator may be written: : , D^n, D^n_x, or \partial_x^n. The derivative of a function ''f'' of an
argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialect ...
''x'' is sometimes given as either of the following: : (x) : f'(x). The ''D'' notation's use and creation is credited to Oliver Heaviside, who considered differential operators of the form : \sum_^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_^n \frac. Another differential operator is the Θ operator, or
theta operator In mathematics, the theta operator is a differential operator defined by : \theta = z . 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 I ...
, defined by :\Theta = z . 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_^n x_k \frac. 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 \overleftarrow g = g \cdot \partial_x f :f \overrightarrow g = f \cdot \partial_x g :f \overleftrightarrow g = f \cdot \partial_x g - g \cdot \partial_x f. Such a bidirectional-arrow notation is frequently used for describing the probability current of quantum mechanics.


Del

The differential operator del, also called ''nabla'', is an important vector differential operator. It appears frequently in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
in places like the differential form of
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. Th ...
. In three-dimensional Cartesian coordinates, del is defined as \nabla = \mathbf + \mathbf + \mathbf . Del defines the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
, and is used to calculate the curl, divergence, and Laplacian of various objects.


Adjoint of an operator

Given a linear differential operator T Tu = \sum_^n a_k(x) D^k u the adjoint 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 the scalar product or inner product. This definition therefore depends on the definition of the scalar product.


Formal adjoint in one variable

In the functional space of square-integrable functions on a real interval , the scalar product is defined by \langle f, g \rangle = \int_a^b \overline \,g(x) \,dx , where the line over ''f''(''x'') denotes the
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of ''f''(''x''). If one moreover adds the condition that ''f'' or ''g'' vanishes as x \to a and x \to b, one can also define the adjoint of ''T'' by T^*u = \sum_^n (-1)^k D^k \left \overline 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 the formal adjoint of ''T''. A (formally) self-adjoint operator is an operator equal to its own (formal) adjoint.


Several variables

If Ω is a domain in R''n'', and ''P'' a differential operator on Ω, then the adjoint of ''P'' is defined in ''L''2(Ω) by duality in the analogous manner: :\langle f, P^* g\rangle_ = \langle P f, g\rangle_ for all smooth ''L''2 functions ''f'', ''g''. Since smooth functions are dense in ''L''2, this defines the adjoint on a dense subset of ''L''2: P* is a densely defined operator.


Example

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


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 mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
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 ''D''2 must be differentiable as many times as the application of ''D''1 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 Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, q ...
: :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.


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).


Ring of polynomial differential operators


Ring of univariate polynomial differential operators

If ''R'' is a ring, let R\langle D,X \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\langle D,X\rangle/I. This is a 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 \text I. It supports an analogue of
Euclidean division of polynomials In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common ...
. Differential modules over R /math> (for the standard derivation) can be identified with modules over R\langle D,X\rangle/I.


Ring of multivariate polynomial differential operators

If ''R'' is a ring, let R\langle D_1,\ldots,D_n,X_1,\ldots,X_n\rangle be the non-commutative polynomial ring over ''R'' in the variables D_1,\ldots,D_n,X_1,\ldots,X_n, and ''I'' the two-sided ideal generated by the elements :(D_i X_j-X_j D_i)-\delta_,\ \ \ D_i D_j -D_j D_i,\ \ \ X_i X_j - X_j X_i for all 1 \le i,j \le n, where \delta is Kronecker delta. Then the ring of multivariate polynomial differential operators over ''R'' is the quotient ring This is a simple ring. Every element can be written in a unique way as a ''R''-linear combination of monomials of the form


Coordinate-independent description

In differential geometry and algebraic geometry it is often convenient to have a
coordinate In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is si ...
-independent description of differential operators between two
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to ev ...
s. Let ''E'' and ''F'' be two vector bundles over a differentiable manifold ''M''. An R-linear mapping of sections is said to be a ''k''th-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) \to F such that :P = i_P\circ j^k where 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 ''k''th-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 In mathematics, the (linear) Peetre theorem, named after Jaak Peetre, is a result of functional analysis that gives a characterisation of differential operators in terms of their effect on generalized function spaces, and without mentioning differe ...
showing that the converse is also true: any (linear) local operator is differential.


Relation to commutative algebra

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'' + 1 smooth functions f_0,\ldots,f_k \in C^\infty(M) we have : _k,[f_,[\cdots[f_0,Pcdots">_,[\cdots[f_0,P.html" ;"title="_k,[f_,[\cdots[f_0,P">_k,[f_,[\cdots[f_0,Pcdots=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 between modules over a
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promi ...
, allowing the concept to be seen as a part of
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promi ...
.


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 In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
, 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 Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promi ...
. 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: \frac = \frac \left( \frac - i \frac \right) \ ,\quad \frac= \frac \left( \frac + i \frac \right) \ . This approach is also used to study functions of several complex variables and functions of a motor variable.


History

The conceptual step of writing a differential operator as something free-standing is attributed to
Louis François Antoine Arbogast Louis François Antoine Arbogast (4 October 1759 – 8 April 1803) was a French mathematician. He was born at Mutzig in Alsace and died at Strasbourg, where he was professor. He wrote on series and the derivatives known by his name: he was the ...
in 1800.James Gasser (editor), ''A Boole Anthology: Recent and classical studies in the logic of George Boole'' (2000), p. 169
Google Books


See also

* Difference operator * Delta operator * Elliptic operator * Curl (mathematics) * Fractional calculus *
Invariant differential operator In mathematics and theoretical physics, an invariant differential operator is a kind of mathematical map from some objects to an object of similar type. These objects are typically functions on \mathbb^n, functions on a manifold, vector valued fu ...
* Differential calculus over commutative algebras *
Lagrangian system In mathematics, a Lagrangian system is a pair , consisting of a smooth fiber bundle and a Lagrangian density , which yields the Euler–Lagrange differential operator acting on sections of . In classical mechanics, many dynamical systems are Lagr ...
* Spectral theory * Energy operator * Momentum operator * DBAR operator


References


External links

* * {{Authority control Operator theory Multivariable calculus