Green's function

In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if is a linear differential operator, then

the Green's function is the solution of the equation where is Dirac's delta function;
the solution of the initial-value problem is the convolution

Through the superposition principle, given a linear ordinary differential equation, one can first solve for each, and realizing that, since the source is a sum of delta functions, the solution is a sum of Green's functions as well, by linearity of.
Green's functions are named after the British mathematician George Green, who first developed the concept in the 1820s. In the modern study of linear partial differential equations, Green's functions are studied largely from the point of view of fundamental solutions instead.
Under many-body theory, the term is also used in physics, specifically in quantum field theory, aerodynamics, aeroacoustics, electrodynamics, seismology and statistical field theory, to refer to various types of correlation functions, even those that do not fit the mathematical definition. In quantum field theory, Green's functions take the roles of propagators.

Definition and uses

A Green's function,, of a linear differential operator acting on distributions over a subset of the Euclidean space at a point, is any solution of
where is the Dirac delta function. This property of a Green's function can be exploited to solve differential equations of the form
If the kernel of is non-trivial, then the Green's function is not unique. However, in practice, some combination of symmetry, boundary conditions and/or other externally imposed criteria will give a unique Green's function. Green's functions may be categorized by a Green's function number according to the type of boundary conditions being satisfied. Green's functions are not necessarily functions of a real variable but are generally understood in the sense of distributions.
Green's functions are also useful tools in solving wave equations and diffusion equations. In quantum mechanics, Green's function of the Hamiltonian is a key concept with important links to the concept of density of states.
The Green's function as used in physics is usually defined with the opposite sign, instead. That is,
This definition does not significantly change any of the properties of Green's function due to the evenness of the Dirac delta function.
If the operator is translation invariant, that is, when has constant coefficients with respect to, then the Green's function can be taken to be a convolution kernel, that is,
In this case, Green's function is the same as the impulse response of linear time-invariant system theory.

Motivation

Loosely speaking, if such a function can be found for the operator, then, if we multiply for the Green's function by, and then integrate with respect to, we obtain,
Because the operator is linear and acts only on the variable , one may take the operator outside of the integration, yielding
This means that
is a solution to the equation
Thus, one may obtain the function through knowledge of the Green's function in and the source term on the right-hand side in. This process relies upon the linearity of the operator.
In other words, the solution of,, can be determined by the integration given in. Although is known, this integration cannot be performed unless is also known. The problem now lies in finding the Green's function that satisfies. For this reason, the Green's function is also sometimes called the fundamental solution associated to the operator.
Not every operator admits a Green's function. A Green's function can also be thought of as a right inverse of. Aside from the difficulties of finding a Green's function for a particular operator, the integral in may be quite difficult to evaluate. However the method gives a theoretically exact result.
This can be thought of as an expansion of according to a Dirac delta function basis (projecting over and a superposition of the solution on each projection. Such an integral equation is known as a Fredholm integral equation, the study of which constitutes Fredholm theory.

Green's functions for solving non-homogeneous boundary value problems

The primary use of Green's functions in mathematics is to solve non-homogeneous boundary value problems. In modern theoretical physics, Green's functions are also usually used as propagators in Feynman diagrams; the term Green's function is often further used for any correlation function.

Framework

Let be the Sturm–Liouville operator, a linear differential operator of the form
and let be the vector-valued boundary conditions operator
Let be a continuous function in Further suppose that the problem
is "regular", i.e., the only solution for for all is

Theorem

There is one and only one solution that satisfies
and it is given by
where is a Green's function satisfying the following conditions:

is continuous in and.
For
For
Derivative "jump":
Symmetry:
Advanced and retarded Green's functions

Green's function is not necessarily unique since the addition of any solution of the homogeneous equation to one Green's function results in another Green's function. Therefore, if the homogeneous equation has nontrivial solutions, multiple Green's functions exist. Certain boundary value or initial value problems involve finding a Green's function that is nonvanishing only for ; in this case, the solution is sometimes called a retarded Green's function. Similarly, a Green's function that is nonvanishing only for is called an advanced Green's function. In such cases, any linear combination of the two Green's functions is also a valid Green's function. Both the advanced and retarded Green's functions are called one-sided, while a Green's function that is nonvanishing for all in the domain of definition is called two-sided.
The terminology advanced and retarded is especially useful when the variable x corresponds to time. In such cases, the solution provided by the use of the retarded Green's function depends only on the past sources and is causal whereas the solution provided by the use of the advanced Green's function depends only on the future sources and is acausal. In these problems, it is often the case that the causal solution is the physically important one. However, the advanced Green's function is useful in finding solutions to certain inverse problems where sources are to be found from boundary data. The use of advanced and retarded Green's function is especially common for the analysis of solutions of the inhomogeneous electromagnetic wave equation.

Finding Green's functions

Eigenvalue expansions

If a differential operator admits a set of eigenvectors that is complete, then it is possible to construct a Green's function from these eigenvectors and eigenvalues.
"Complete" means that the set of functions satisfies the following completeness relation,
Then the following holds,
where represents complex conjugation.
Applying the operator to each side of this equation results in the completeness relation, which was assumed.
The general study of Green's function written in the above form, and its relationship to the function spaces formed by the eigenvectors, is known as Fredholm theory.
There are several other methods for finding Green's functions, including the method of images, separation of variables, and Laplace transforms.

Representations in terms of the Wronskian

Let be the general linear second-order differential operator defined on. We write
Suppose that and together form a basis of linearly independent solutions to the homogeneous problem Given homogeneous boundary conditions for the Green's function, we may construct by requiring and The Green's function satisfying these conditions, alongside the continuity of and its derivative "jump", can be written as
where is known as the Wronskian determinant of and. Though this is a somewhat limited case, the Wronskian frequently appears in other sets of boundary value problems that require a one-sided Green's function as well, including those with conditions on boundary derivatives or a pair of conditions on a function and its normal derivative on a single boundary.

Combining Green's functions

If the differential operator can be factored as then the Green's function of can be constructed from the Green's functions for and
The above identity follows immediately from taking to be the representation of the right operator inverse of analogous to how for the invertible linear operator defined by is represented by its matrix elements
A further identity follows for differential operators that are scalar polynomials of the derivative, The fundamental theorem of algebra, combined with the fact that commutes with itself, guarantees that the polynomial can be factored, putting in the form:
where are the zeros of Taking the Fourier transform of with respect to both and gives:
The fraction can then be split into a sum using a partial fraction decomposition before Fourier transforming back to and space. This process yields identities that relate integrals of Green's functions and sums of the same. For example, if then one form for its Green's function is:
While the example presented is tractable analytically, it illustrates a process that works when the integral is not trivial.

Table of Green's functions

The following table gives an overview of Green's functions of frequently appearing differential operators, where is the Heaviside step function, is a Bessel function, is a modified Bessel function of the first kind, and is a modified Bessel function of the second kind. Where time appears in the first column, the retarded Green's function is listed.

Differential operator	Green's function	Example of application



where	with	1D underdamped harmonic oscillator
where	with	1D overdamped harmonic oscillator
where		1D critically damped harmonic oscillator
1D Laplace operator		1D Poisson equation
2D Laplace operator	with	2D Poisson equation
3D Laplace operator	with	Poisson equation
Helmholtz operator	where is the Hankel function of the second kind, and is the spherical Hankel function of the second kind	stationary 3D Schrödinger equation for free particle
	where is the Hankel function of the first kind, and is the modified Bessel function	2D time-harmonic flexural wave equation
Divergence operator		Let be a vector field from to, and with, such that. The function,, is the gamma function, and is Kronecker's delta, such that for, and for. Lastly, ! is the factorial symbol and is the absolute value.
in dimensions		Yukawa potential, Feynman propagator, Screened Poisson equation
		1D wave equation
		2D wave equation
D'Alembert operator		3D wave equation
		1D diffusion
		2D diffusion
		3D diffusion
	with	1D Klein–Gordon equation
	with	2D Klein–Gordon equation
	with	3D Klein–Gordon equation
	with and	telegrapher's equation
	with and	2D relativistic heat conduction
	with and	3D relativistic heat conduction