Differential operator

Summary

In mathematics, a differential operator is an 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 that accepts a function and returns another function (in the style of a higher-order function in computer science).

A harmonic function defined on an annulus. Harmonic functions are exactly those functions which lie in the kernel of the Laplace operator, an important differential operator.

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 edit

Given a nonnegative integer m, an order-  linear differential operator is a map   from a function space   to another function space   that can be written as:

 
where   is a multi-index of non-negative integers,  , and for each  ,   is a function on some open domain in n-dimensional space. The operator   is interpreted as

 

Thus for a function  :

 

The notation   is justified (i.e., independent of order of differentiation) because of the symmetry of second derivatives.

The polynomial p obtained by replacing D by variables   in P is called the total symbol of P; i.e., the total symbol of P above is:

 
where   The highest homogeneous component of the symbol, namely,
 

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

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

 

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

 

where, for each multi-index α,   is a bundle map, symmetric on the indices α.

The kth order coefficients of P transform as a symmetric tensor

 

whose domain is the tensor product of the kth 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 ξi. In terms of a basis of frames eμ, fν of E and F, respectively, the differential operator P decomposes into components

 

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

 

With this trivialization, the principal symbol can now be written

 

In the cotangent space over a fixed point x of X, the symbol   defines a homogeneous polynomial of degree k in   with values in  .

Fourier interpretation edit

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,

 

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.

Examples edit

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

History edit

The conceptual step of writing a differential operator as something free-standing is attributed to Louis François Antoine Arbogast in 1800.[2]

Notations edit

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:

 ,  ,   and  .

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

 ,  ,  , or  .

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

 
 

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

 

in his study of differential equations.

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

 

Another differential operator is the Θ operator, or theta operator, defined by[3]

 

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

 

In n variables the homogeneity operator is given by

 

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:

 
 
 

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

Adjoint of an operator edit

Given a linear differential operator  

 
the adjoint of this operator is defined as the operator   such that
 
where the notation   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 edit

In the functional space of square-integrable functions on a real interval (a, b), the scalar product is defined by

 

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   and  , one can also define the adjoint of T by

 

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   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 edit

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:

 

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.

Example edit

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

 

This property can be proven using the formal adjoint definition above.[4]

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

Properties of differential operators edit

Differentiation is linear, i.e.

 
 

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

 

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:

 

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 edit

Ring of univariate polynomial differential operators edit

If R is a ring, let   be the non-commutative polynomial ring over R in the variables D and X, and I the two-sided ideal generated by DXXD − 1. Then the ring of univariate polynomial differential operators over R is the quotient ring  . This is a non-commutative simple ring. Every element can be written in a unique way as a R-linear combination of monomials of the form  . It supports an analogue of Euclidean division of polynomials.

Differential modules[clarification needed] over   (for the standard derivation) can be identified with modules over  .

Ring of multivariate polynomial differential operators edit

If R is a ring, let   be the non-commutative polynomial ring over R in the variables  , and I the two-sided ideal generated by the elements

 

for all   where   is Kronecker delta. Then the ring of multivariate polynomial differential operators over R is the quotient ring  .

This is a non-commutative simple ring. Every element can be written in a unique way as a R-linear combination of monomials of the form  .

Coordinate-independent description edit

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 P : Γ(E) → Γ(F) is said to be a kth-order linear differential operator if it factors through the jet bundle Jk(E). In other words, there exists a linear mapping of vector bundles

 

such that

 

where jk: Γ(E) → Γ(Jk(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.

Relation to commutative algebra edit

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   we have

 

Here the bracket   is defined as the commutator

 

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.

Variants edit

A differential operator of infinite order edit

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

Bidifferential operator edit

A differential operator acting on two functions   is called a bidifferential operator. The notion appears, for instance, in an associative algebra structure on a deformation quantization of a Poisson algebra.[5]

Microdifferential operator edit

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.[6]

See also edit

References edit

  1. ^ Schapira 1985, 1.1.7
  2. ^ James Gasser (editor), A Boole Anthology: Recent and classical studies in the logic of George Boole (2000), p. 169; Google Books.
  3. ^ E. W. Weisstein. "Theta Operator". Retrieved 2009-06-12.
  4. ^
     
  5. ^ 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.
  6. ^ Schapira 1985, § 1.2. § 1.3.
  • 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 edit

  • 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.

External links edit