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 L is the linear differential operator, then
Through the superposition principle, given a linear ordinary differential equation, L = source, 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 1830s. 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 the type of boundary conditions satisfied, by a Green's function number. Also, Green's functions in general are distributions, not necessarily functions of a real variable.
Green's functions are also useful tools in solving wave equations and diffusion equations. In quantum mechanics, the 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 the 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, the 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 the equation 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 equation and the source term on the right-hand side in equation . This process relies upon the linearity of the operator.
In other words, the solution of equation ,, can be determined by the integration given in equation . Although is known, this integration cannot be performed unless is also known. The problem now lies in finding the Green's function that satisfies equation . 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 equation 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 inhomogeneous 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:
  1. is continuous in and.
  2. For,.
  3. For,.
  4. Derivative "jump":.
  5. Symmetry:.

    Advanced and retarded Green's functions

Sometimes the Green's function can be split into a sum of two functions. One with the variable positive and the other with the variable negative. These are the advanced and retarded Green's functions, and when the equation under study depends on time, one of the parts is causal and the other anti-causal. In these problems usually the causal part is the important one. These are frequently the solutions to the inhomogeneous electromagnetic wave equation.

Finding Green's functions

Units

While it doesn't uniquely fix the form the Green's function will take, performing a dimensional analysis to find the units a Green's function must have is an important sanity check on any Green's function found through other means. A quick examination of the defining equation,
shows that the units depend not only on the units of but also on the number and units of the space of which the position vectors and are elements. This leads to the relationship:
where is defined as, "the physical units of ", and is the volume element of the space.
For example, if and time is the only variable then:
If, the d'Alembert operator, and space has 3 dimensions then:

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 the 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.

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. Where time appears in the first column, the advanced Green's function is listed.
Differential operator Green's function Example of application
with 1D damped harmonic oscillator
2D Laplace operator with 2D Poisson equation
3D Laplace operator with Poisson equation
Helmholtz operatorstationary 3D Schrödinger equation for free particle
in dimensionsYukawa potential, Feynman propagator
1D wave equation
2D wave equation
D'Alembert operator3D wave equation
1D diffusion
2D diffusion
3D diffusion
1D Klein–Gordon equation
2D Klein–Gordon equation
3D Klein–Gordon equation
telegrapher's equation
2D relativistic heat conduction
3D relativistic heat conduction

Green's functions for the Laplacian

Green's functions for linear differential operators involving the Laplacian may be readily put to use using the second of Green's identities.
To derive Green's theorem, begin with the divergence theorem,
Let and substitute into Gauss' law.
Compute and apply the product rule for the ∇ operator,
Plugging this into the divergence theorem produces Green's theorem,
Suppose that the linear differential operator is the Laplacian, ∇², and that there is a Green's function for the Laplacian. The defining property of the Green's function still holds,
Let in Green's second identity, see Green's identities. Then,
Using this expression, it is possible to solve Laplace's equation ∇2φ = 0 or Poisson's equation ∇2φ = −ρ, subject to either Neumann or Dirichlet boundary conditions. In other words, we can solve for φ everywhere inside a volume where either the value of φ is specified on the bounding surface of the volume, or the normal derivative of φ is specified on the bounding surface.
Suppose the problem is to solve for φ inside the region. Then the integral
reduces to simply φ due to the defining property of the Dirac delta function and we have
This form expresses the well-known property of harmonic functions, that if the value or normal derivative is known on a bounding surface, then the value of the function inside the volume is known everywhere.
In electrostatics, φ is interpreted as the electric potential, ρ as electric charge density, and the normal derivative as the normal component of the electric field.
If the problem is to solve a Dirichlet boundary value problem, the Green's function should be chosen such that G vanishes when either x or x′ is on the bounding surface. Thus only one of the two terms in the surface integral remains. If the problem is to solve a Neumann boundary value problem, the Green's function is chosen such that its normal derivative vanishes on the bounding surface, as it would seem to be the most logical choice.. However, application of Gauss's theorem to the differential equation defining the Green's function yields
meaning the normal derivative of G cannot vanish on the surface, because it must integrate to 1 on the surface. .
The simplest form the normal derivative can take is that of a constant, namely 1/S, where S is the surface area of the surface. The surface term in the solution becomes
where is the average value of the potential on the surface. This number is not known in general, but is often unimportant, as the goal is often to obtain the electric field given by the gradient of the potential, rather than the potential itself.
With no boundary conditions, the Green's function for the Laplacian is
Supposing that the bounding surface goes out to infinity and plugging in this expression for the Green's function finally yields the standard expression for electric potential in terms of electric charge density as

Example

Example. Find the Green function for the following problem, whose Green's function number is X11:
First step: The Green's function for the linear operator at hand is defined as the solution to
If, then the delta function gives zero, and the general solution is
For, the boundary condition at implies
if and.
For, the boundary condition at implies
The equation of is skipped for similar reasons.
To summarize the results thus far:
Second step: The next task is to determine and.
Ensuring continuity in the Green's function at implies
One can ensure proper discontinuity in the first derivative by integrating the defining differential equation from to and taking the limit as goes to zero:
The two continuity equations can be solved for and to obtain
So the Green's function for this problem is:

Further examples

  • Let = 1 and let the subset be all of ℝ. Let be. Then, the Heaviside step function is a Green's function of at 0.
  • Let = 2 and let the subset be the quarter-plane and be the Laplacian. Also, assume a Dirichlet boundary condition is imposed at = 0 and a Neumann boundary condition is imposed at = 0. Then the X10Y20 Green's function is
  • Let, and all three are elements of the real numbers. Then, for any function from reals to reals,, with an th derivative that is integrable over the interval :

    Footnotes