In mathematics, a Green’s function is a type of function used to solve inhomogeneous differential equations subject to boundary conditions. The term is also used in physics, specifically in quantum field theory, electrodynamics and statistical field theory, to refer to various types of correlation functions, even those that do not fit the mathematical definition; for this sense, see Correlation function (quantum field theory) and Green’s function (many-body theory).
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 largely studied from the point of view of fundamental solutions instead.
Definition and uses
Technically, 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 technique can be used to solve differential equations of the form
If the kernel of is nontrivial, 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. Also, Green’s functions in general are distributions, not necessarily proper functions.
Green’s functions are also a useful tool in condensed matter theory, where they allow the resolution of the diffusion equation, and in quantum mechanics, where the Green’s function of the Hamiltonian is a key concept, with important links to the concept of density of states. The Green’s functions used in those two domains are highly similar, due to the analogy in the mathematical structure of the diffusion equation and Schrödinger equation. As a side note, the Green’s function as used in Physics is usually defined with the opposite sign; that is, . This definition does not significantly change any of the properties of the Green’s function.
Loosely speaking, if such a function can be found for the operator , then if we multiply the equation (1) for the Green’s function by , and then perform an integration in the variable, we obtain
The right hand side is now given by the equation (2) to be equal to , thus
Because the operator is linear and acts on the variable alone (not on the variable of integration ), we can take the operator outside of the integration on the right hand side obtaining
And this implies
Thus, we can obtain the function through knowledge of the Green’s function in equation (1), and the source term on the right hand side in equation (2). This process has resulted from the linearity of the operator .
In other words, the solution of equation (2), , can be determined by the integration given in equation (3). 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 (1). 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 functions for a particular operator, the integral in equation (3), may be quite difficult to perform. 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 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 inhomogeneous boundary value problems. In modern theoretical physics, Green’s functions are also usually used as propagators in Feynman diagrams (and the phrase “Green’s function” is often used for any correlation function).
Let be the Sturm-Liouville operator, a linear differential operator of the form
and let be the boundary conditions operator
Let be a continuous function in . We shall also suppose that the problem
is regular, i.e. only the trivial solution exists for the homogeneous problem.
Theorem. There is one and only one solution which 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: .
Finding Green’s functions (eigenvalue expansions)
If a differential operator admits a set of eigenvectors (i.e. a set of functions and scalars such that ) that are 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
Applying the operator to each side of this equation results in the completeness relation, which was assumed true.
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.
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 (otherwise known as Gauss’s law)
Let and substitute into Gauss’ law. Compute $latex \nabla\cdot\hat A$ and apply the chain rule for the operator
Plugging this into the divergence theorem produces Green’s theorem
Suppose that the linear differential operator is the Laplacian, , 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 theorem. Then
Using this expression, it is possible to solve Laplace’s equation or Poisson’s equation , subject to either Neumann or Dirichlet boundary conditions. In other words, we can solve for everywhere inside a volume where either (1) the value of is specified on the bounding surface of the volume (Dirichlet boundary conditions), or (2) the normal derivative of is specified on the bounding surface (Neumann boundary conditions).
Suppose the problem is to solve for inside the region. Then the integral
reduces to simply 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, is interpreted as the electric potential, 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 vanishes when either or is on the bounding surface; conversely, 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. Thus only one of the two terms in the surface integral remains.
With no boundary conditions, the Green’s function for the Laplacian (Green’s function for the three-variable Laplace equation) is
Supposing that the bounding surface goes out to infinity, and plugging in this expression for the Green’s function, this gives the familiar expression for electric potential in terms of electric charge density (in the CGS unit system) as
Example. Given the problem
Find the Green’s function.
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 which implies . The equation of is skipped because if and . For , the boundary condition at implies which 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 also 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 (dis)continuity equations can be solved for and to obtain
So the Green’s function for this problem is
- Let and let the subset be all of . Let be . Then, the Heaviside step function is a Green’s function of at .
- Let and let the subset be the quarter-plane and be the Laplacian. Also, assume a Dirichlet boundary condition is imposed at and a Neumann boundary condition is imposed at . Then the Green’s function is
which completes this example.