In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to boundary conditions. Technically, a Green's function of a linear operator L acting on distributions over a manifold M, at a point x₀, is any solution of (Lf)(x) = δ(x − x₀), where δ is the Dirac delta function. If the kernel of L 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 would give us a unique Green's function. Also, please note that Green's functions in general are distributions, not necessarily proper functions.

Not every operator L admits a Green's function. A Green's function can also be thought of as a one-sided inverse of L.

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.

The Green's function was named after British mathematician George Green, who first developed the concept in the 1830s.

Motivation

---

Convolving with a Green's function gives solutions to inhomogeneous differential-integral equations, most commonly a Sturm-Liouville problem. If g is the Green's function of an operator L, then the solution for f of the equation Lf = h is given by

$f(x)\; =\; \backslash int\{\; h(s)\; g(x,s)\; \backslash ,\; ds\}.$

This can be thought of as an expansion of h according to a Dirac delta function basis (projecting h over δ(x − s)) 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 function for solving inhomogeneous boundary value problems

---

The primary use of Green's functions in mathematics is to solve inhomogeneous boundary value problems. In particle 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).

Working frame

Let $L$ be the Sturm-Liouville operator, a linear differential operator of the form

$L\; =\; \{d\; \backslash over\; dx\}\backslash leftp(x)\; \{d\; \backslash over\; dx\}\; \backslash right+\; q(x)$

and let D be the boundary conditions operator

$Du\; =\; \backslash left\backslash \{\backslash begin\{matrix\}\; \backslash alpha\; \_1\; u\text{'}(0)\; +\; \backslash beta\; \_1\; u(0)\; \backslash \backslash \; \backslash alpha\; \_2\; u\text{'}(l)\; +\; \backslash beta\; \_2\; u(l)\; \backslash end\{matrix\}\backslash right.$

Let $f(x)$ be a continuous function in $*$. We shall also suppose that the problem

$\backslash begin\{matrix\}Lu\; =\; f\; \backslash \backslash \; Du\; =\; 0\; \backslash end\{matrix\}$

is regular, i.e. only the trivial solution exists for the homogeneous problem.

Theorem

Then there is one and only one solution u(x) which satisfies

$\backslash begin\{matrix\}Lu\; =\; f\; \backslash \backslash \; Du\; =\; 0\; \backslash end\{matrix\}$

and it is given by

$u(x)\; =\; \backslash int\_\{0\}^\{l\}\{\; f(s)\; g(x,s)\; \backslash ,\; ds\}$

where g(x,s) is Green's function and satisfies the following demands:

1. g(x,s) is continuous in x and s.
2. For $x\; \backslash ne\; s$, $L\; g\; (\; x,\; s\; )\; =\; 0$.
3. For $s\; \backslash ne\; 0,\; l$, $D\; g\; (\; x,\; s\; )\; =\; 0$.
4. Derivative "jump": $g\; \text{'}\; (\; s\_\{\; +\; 0\},\; s\; )\; -\; g\; \text{'}\; (s\_\{\; -\; 0\},\; s\; )\; =\; 1\; /\; p(s)$.
5. Symmetry: g(x, s) = g(s, x).

Finding Green's functions

---

Eigenvalue expansions

If a differential operator L admits a set of eigenvectors $\backslash Psi\_n(x)$ (i.e. a set of functions $\backslash Psi\_n(x)$ and scalars $\backslash lambda\_n$ such that $L\; \backslash Psi\_n\; =\; \backslash lambda\_n\; \backslash Psi\_n)$) that are complete, then we can construct a Green's function from these eigenvectors and eigenvalues.

By complete, we mean that the set of functions :$\backslash Psi\_n(x)$ satisfies the following completeness relation:

$\backslash delta(x\; -\; x\text{'})\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash Psi\_n(x)\; \backslash Psi\_n(x\text{'}).$

We can prove the following:

$G(x,\; x\text{'})\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{\backslash Psi\_n(x)\; \backslash Psi\_n(x\text{'})\}\{\backslash lambda\_n\}.$

Now consider acting on this on each side with the operator L. We'll end up with 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 function 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):

$\backslash int\_V\; \backslash nabla\; \backslash cdot\; \backslash hat\; A\backslash \; dV\; =\; \backslash int\_S\; \backslash hat\; A\; \backslash cdot\; d\backslash hat\backslash sigma$

Let $A\; =\; \backslash phi\backslash nabla\backslash psi\; -\; \backslash psi\backslash nabla\backslash phi$ and substitute into Gauss' law. Compute $\backslash nabla\backslash cdot\backslash hat\; A$ and apply the chain rule for the $\backslash nabla$ operator:

$\backslash nabla\backslash cdot\backslash hat\; A\; =\; \backslash nabla\backslash cdot(\backslash phi\backslash nabla\backslash psi\; -\; \backslash psi\backslash nabla\backslash phi)\; =\; (\backslash nabla\backslash phi)\backslash cdot(\backslash nabla\backslash psi)\; +\; \backslash phi\backslash nabla^2\backslash psi\; -\; (\backslash nabla\backslash phi)\backslash cdot(\backslash nabla\backslash psi)\; -\; \backslash psi\backslash nabla^2\backslash phi\; =\; \backslash phi\backslash nabla^2\backslash psi\; -\; \backslash psi\backslash nabla^2\backslash phi$

Plugging this into the divergence theorem, we arrive at Green's theorem:

$\backslash int\_V\; \backslash phi\backslash nabla^2\backslash psi\; -\; \backslash psi\backslash nabla^2\backslash phi\backslash \; dV\; =\; \backslash int\_S\; \backslash phi\backslash nabla\backslash psi\; -\; \backslash psi\backslash nabla\backslash phi\; \backslash cdot\; d\backslash hat\backslash sigma$

Suppose that our linear differential operator L is the laplacian, $\backslash nabla^2$, and that we have a Green's function G for the laplacian. The defining property of the Green's function still holds:

$L\; G(x,x\text{'})\; =\; \backslash nabla^2\; G(x,x\text{'})\; =\; \backslash delta(x-x\text{'})$

Let $\backslash psi\; =\; G$ in Green's theorem. We get:

$\backslash int\_V\; \backslash phi(x\text{'})\; \backslash delta(x\; -\; x\text{'})\; -\; G(x,x\text{'})\; \backslash nabla^2\backslash phi(x\text{'})\backslash \; d^3x\text{'}\; =\; \backslash int\_S\; \backslash phi(x\text{'})\backslash nabla\text{'}\; G(x,x\text{'})\; -\; G(x,x\text{'})\backslash nabla\text{'}\backslash phi(x\text{'})\; \backslash cdot\; d\backslash hat\backslash sigma\text{'}$

Using this expression, we can solve Laplace's equation $\backslash nabla^2\backslash phi(x)=0$ or Poisson's equation $\backslash nabla^2\backslash phi(x)=-4\backslash pi\backslash rho(x)$, subject to either Neumann or Dirichlet boundary conditions. In other words, we can solve for $\backslash phi(x)$ everywhere inside a volume where either (1) the value of $\backslash phi(x)$ is specified on the bounding surface of the volume (Dirichlet boundary conditions), or (2) the normal derivative of $\backslash phi(x)$ is specified on the bounding surface.

Suppose we're interested in solving for $\backslash phi(x)$ inside the region. Then the integral

$\backslash int\backslash limits\_V\; \{\backslash phi(x\text{'})\backslash delta(x-x\text{'})\backslash \; d^3x\text{'}\}$

reduces to simply $\backslash phi(x)$ due to the defining property of the Dirac delta function and we have:

$\backslash phi(x)\; =\; \backslash int\_V\; G(x,x\text{'})\; \backslash rho(x\text{'})\backslash \; d^3x\text{'}\; +\; \backslash int\_S\; \backslash phi(x\text{'})\backslash nabla\text{'}\; G(x,x\text{'})\; -\; G(x,x\text{'})\backslash nabla\text{'}\backslash phi(x\text{'})\; \backslash cdot\; d\backslash hat\backslash sigma\text{'}$

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, we interpret $\backslash phi(x)$ as the electric potential, $\backslash rho(x)$ as electric charge density, and the normal derivative $\backslash nabla\backslash phi(x\text{'})\backslash cdot\; d\backslash hat\backslash sigma\text{'}$ as the normal component of the electric field.

If we're interested in solving a Dirichlet boundary value problem, we choose our Green's function such that $G(x,x\text{'})$ vanishes when either x or x' is on the bounding surface; conversely, if we're interested in solving a Neumann boundary value problem, we choose our Green's function such that its normal derivative vanishes on the bounding surface. Thus we are left with only one of the two terms in the surface integral.

With no boundary conditions, the Green's function for the Laplacian is:

$G(\backslash hat\; x,\; \backslash hat\; x\text{'})\; =\; \backslash frac\{1\}\{|\backslash hat\; x\; -\; \backslash hat\; x\text{'}|\}$

Supposing that our bounding surface goes out to infinity, and plugging in this expression for the Green's function, we arrive at the familiar expression for electric potential in terms of electric charge density:

$\backslash phi(x)\; =\; \backslash int\_V\; \backslash frac\{\backslash rho(x\text{'})\}\{|\backslash hat\; x\; -\; \backslash hat\; x\text{'}|\}\; \backslash \; d^3x\text{'}$

Example

---

Given the problem

$\backslash begin\{matrix\}Lu\backslash end\{matrix\}\; =\; u\; \text{'}\; \text{'}\; +\; u\; =\; f(\; x\; )$

$Du\; =\; u(0)\; =\; 0\; \backslash quad,\; \backslash quad\; u\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash right)\; =\; 0$

Find Green's function.

First step: From demand-2 we see that

$g(x,s)\; =\; c\_1\; (s)\; \backslash cdot\; \backslash cos\; x\; +\; c\_2\; (s)\; \backslash cdot\; \backslash sin\; x$

For x < s we see from demand-3 that the $c\_1\; (s)\; =\; 0$, while for x > s we see from demand-3 that the $c\_2\; (s)\; =\; 0$ (we leave it to the reader to fill in the in-between steps).

Summarize the results:

$g(x,s)=\backslash left\backslash \{\backslash begin\{matrix\}$

a(s) \sin x, \;\; x < s \\ b(s) \cos x, \;\; s < x \end{matrix}\right.

Second step: Now we shall determine a(s) and b(s).

Using demand-1 we get

$a(s)\; \backslash sin\; s\; =\; b(s)\; \backslash cos\; s\backslash quad$.

Using demand-4 we get

$b(s)\; \backslash cdot-\; \backslash sin\; s-\; a(s)\; \backslash cdot\; \backslash cos\; s\; =\; \backslash frac\{1\}\{1\}\; =\; 1\backslash ,\; .$

Using Cramer's rule or by intelligent guess solve for a(s) and b(s) and obtain that $a(s)\; =\; -\; \backslash cos\; s\; \backslash quad\; ;\; \backslash quad\; b(s)\; =\; -\; \backslash sin\; s$.

Check that this automatically satisfies demand-5.

So our Green's function for this problem is:

$g(x,s)=\backslash left\backslash \{\backslash begin\{matrix\}$

-1 \cdot \cos s \cdot \sin x, \;\; x < s \\ -1 \cdot \sin s \cdot \cos x, \;\; s < x \end{matrix}\right.

Green's functions in condensed matter physics

---

Green's functions are widely used in condensed matter physics. For a system of non-interacting particles described by a Hamiltonian $H$, we can define the Green's function $g(E)$ at energy $E$ as

$g^\{\backslash pm\}(E)\; =\; (E\; \backslash pm\; i0^+-\; H)^\{-1\},$

where the +(-) signifies the retarded (advanced) Green's function.

Then the density of one particle states at energy E is written as

$\backslash rho(E)=\backslash frac\{-1\}\{\backslash pi\}\backslash int\; dx\; \backslash textrm\{Im\}(\backslash langle\; x|g(E)|x\; \backslash rangle)).$

Further examples

---

• Let the manifold be R and L be d/dx. Then, the Heaviside step function H(x − x₀) is a Green's function of L at x₀.
• Let the manifold be the quarter-plane { (x, y) : x, y ≥ 0 } and L be the Laplacian. Also, assume a Dirichlet boundary condition is imposed at x = 0 and a Neumann boundary condition is imposed at y = 0. Then the Green's function is

$G(x,\; y,\; x\_0,\; y\_0)=\backslash frac\{1\}\{2\backslash pi\}\backslash left*$

$+\backslash frac\{1\}\{2\backslash pi\}\backslash left*.$

See also

---

• Discrete Green's functions can be defined on graphs and grids.
• impulse response

Bibliography

---

• Eyges, Leonard, The Classical Electromagnetic Field, Dover Publications, New York, 1972. ISBN 0-486-63947-9. (Chapter 5 contains a very readable account of using Green's functions to solve boundary value problems in electrostatics.)
• A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition), Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2
• A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9

External links

---

Differential equations

Greensche Funktion | Fonction de Green | グリーン関数 | Функция Грина