In mathematics, a Green's function (or Green 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 {\displaystyle L} is a linear differential operator, then
the Green's function G {\displaystyle G} is the solution of the equation L G = δ , {\displaystyle LG=\delta ,} where δ {\displaystyle \delta } is Dirac's delta function; the solution of the inhomogeneous problem L y = f {\displaystyle Ly=f} is the convolution, y = ( G ∗ f ) . {\displaystyle y=(G\ast f).}
By the superposition principle, given a linear ordinary differential equation (ODE), L y = f {\displaystyle Ly=f} , one can first solve L G = δ s {\displaystyle LG=\delta _{s}} , for each s. If the source is a sum of delta functions, then the solution is a sum of Green's functions as well due to linearity of L. This means that the integral, viewed as a continuous sum, can reconstruct a wide class of sources, f {\displaystyle f} , through the convolution integral. Whenever the integral of f {\displaystyle f} with G {\displaystyle G} converges, then the solution to the inhomogeneous equation, L y = f {\displaystyle Ly=f} , is given by y = G ∗ f {\displaystyle y=G\ast f} . 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, which take into account the modern language of the theory of distributions or generalized functions. Building off of the superposition principle in many-body theory, the term is also used in physics and engineering, 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 role of propagators, also referred to as two-point (correlation) functions.
Definition and uses A Green's function, G(x,s), of a linear differential operator L = L(x) acting on distributions over a subset of the Euclidean space R n {\displaystyle \mathbb {R} ^{n}} , at a point s, is any solution of
where δ 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 L 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,
L G ( x , s ) = δ ( x − s ) . {\displaystyle LG(x,s)=\delta (x-s)\,.}
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 L {\displaystyle L} has constant coefficients with respect to x, then the Green's function can be taken to be a convolution kernel, that is,
G ( x , s ) = G ( x − s ) . {\displaystyle G(x,s)=G(x-s)\,.}
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 G can be found for the operator L, then, if we multiply equation 1 for the Green's function by f(s), and then integrate with respect to s, we obtain,
∫ L G ( x , s ) f ( s ) d s = ∫ δ ( x − s ) f ( s ) d s = f ( x ) . {\displaystyle \int LG(x,s)\,f(s)\,ds=\int \delta (x-s)\,f(s)\,ds=f(x)\,.}
Because the operator L = L ( x ) {\displaystyle L=L(x)} is linear and acts only on the variable x (and not on the variable of integration s), one may take the operator L {\displaystyle L} outside of the integration, yielding
L ( ∫ G ( x , s ) f ( s ) d s ) = f ( x ) . {\displaystyle L\left(\int G(x,s)\,f(s)\,ds\right)=f(x)\,.}
This means that
is a solution to the equation L u ( x ) = f ( x ) . {\displaystyle Lu(x)=f(x)\,.}
Thus, one may obtain the function u(x) through knowledge of the Green's function in equation 1 and the source term on the right-hand side in equation 2. This process relies upon the linearity of the operator L. In other words, the solution of equation 2, u(x), can be determined by the integration given in equation 3. Although f(x) is known, this integration cannot be performed unless G is also known. The problem now lies in finding the Green's function G that satisfies equation 1. For this reason, the Green's function is also sometimes called the fundamental solution associated to the operator L. Not every operator L {\displaystyle L} admits a Green's function. A Green's function can also be thought of as a right inverse of L. Aside from the difficulties of finding a Green's function for a particular operator, the integral in equation 3 may be quite difficult to evaluate. However the method gives a theoretically exact result. This can be thought of as an expansion of f according to a Dirac delta function basis (projecting f over δ ( x − s ) {\displaystyle \delta (x-s)} ; 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 L {\displaystyle L} be a Sturm–Liouville operator, a linear differential operator of the form
L = d d x [ p ( x ) d d x ] + q ( x ) {\displaystyle L={\dfrac {d}{dx}}\left[p(x){\dfrac {d}{dx}}\right]+q(x)}
and let D {\displaystyle \mathbf {D} } be the vector-valued boundary conditions operator
D u = [ α 1 u ′ ( 0 ) + β 1 u ( 0 ) α 2 u ′ ( ℓ ) + β 2 u ( ℓ ) ] . {\displaystyle \mathbf {D} u={\begin{bmatrix}\alpha _{1}u'(0)+\beta _{1}u(0)\\\alpha _{2}u'(\ell )+\beta _{2}u(\ell )\end{bmatrix}}\,.}
Let f ( x ) {\displaystyle f(x)} be a continuous function in [ 0 , ℓ ] {\displaystyle [0,\ell ]\,} . Further suppose that the problem
L u = f D u = 0 {\displaystyle {\begin{aligned}Lu&=f\\\mathbf {D} u&=\mathbf {0} \end{aligned}}}
is regular, i.e., the only solution for f ( x ) = 0 {\displaystyle f(x)=0} for all x is u ( x ) = 0 {\displaystyle u(x)=0} .
Theorem There is one and only one solution u ( x ) {\displaystyle u(x)} that satisfies
L u = f D u = 0 {\displaystyle {\begin{aligned}Lu&=f\\\mathbf {D} u&=\mathbf {0} \end{aligned}}}
and it is given by
u ( x ) = ∫ 0 ℓ f ( s ) G ( x , s ) d s , {\displaystyle u(x)=\int _{0}^{\ell }f(s)\,G(x,s)\,ds\,,}
where G ( x , s ) {\displaystyle G(x,s)} is a Green's function satisfying the following conditions:
G ( x , s ) {\displaystyle G(x,s)} is continuous in x {\displaystyle x} and s {\displaystyle s} . For x ≠ s {\displaystyle x\neq s\,} , L G ( x , s ) = 0 {\displaystyle LG(x,s)=0} . For s ≠ 0 {\displaystyle s\neq 0\,} , D G ( x , s ) = 0 {\displaystyle \mathbf {D} G(x,s)=\mathbf {0} } . Derivative "jump": G ′ ( s 0 + , s ) − G ′ ( s 0 − , s ) = 1 / p ( s ) {\displaystyle G'(s_{0+},s)-G'(s_{0-},s)=1/p(s)\,} . Symmetry: G ( x , s ) = G ( s , x ) {\displaystyle G(x,s)=G(s,x)\,} .
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 s ≤ x {\displaystyle s\leq x} ; in this case, the solution is sometimes called a retarded Green's function. Similarly, a Green's function that is nonvanishing only for s ≥ x {\displaystyle s\geq x} 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 x {\displaystyle x} 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 L admits a set of eigenvectors Ψn(x) (i.e., a set of functions Ψn and scalars λn such that LΨn = λn Ψn ) 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 {Ψn} satisfies the following completeness relation,
δ ( x − x ′ ) = ∑ n = 0 ∞ Ψ n † ( x ′ ) Ψ n ( x ) . {\displaystyle \delta (x-x')=\sum _{n=0}^{\infty }\Psi _{n}^{\dagger }(x')\Psi _{n}(x).}
Then the following holds,
where † {\displaystyle \dagger } represents complex conjugation. Applying the operator L 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 L {\displaystyle L} be the general linear second-order differential operator defined on [ a , b ] ∈ R {\displaystyle [a,b]\in \mathbb {R} } . We write
L u ( x ) = α ( x ) d 2 d x 2 u ( x ) + β ( x ) d d x u ( x ) + γ ( x ) u ( x ) = f ( x ) . {\displaystyle Lu(x)=\alpha (x){\frac {d^{2}}{dx^{2}}}u(x)+\beta (x){\frac {d}{dx}}u(x)+\gamma (x)u(x)=f(x).}
Suppose that u 1 {\displaystyle u_{1}} and u 2 {\displaystyle u_{2}} together form a basis of linearly independent solutions to the homogeneous problem L u = 0. {\displaystyle Lu=0.} Given homogeneous boundary conditions for the Green's function G ( a , s ) = G ( b , s ) = 0 {\displaystyle G(a,s)=G(b,s)=0} , we may construct G ( x , s ) {\displaystyle G(x,s)} by requiring u 1 ( a ) = 0 {\displaystyle u_{1}(a)=0} and u 2 ( b ) = 0. {\displaystyle u_{2}(b)=0.} The Green's function satisfying these conditions, alongside the continuity of G {\displaystyle G} and its derivative "jump", can be written as
G ( x , s ) = { u 1 ( x ) u 2 ( s ) α ( s ) W ( s ) , a ≤ x < s u 2 ( x ) u 1 ( s ) α ( s ) W ( s ) , s < x ≤ b {\displaystyle G(x,s)={\begin{cases}{\dfrac {u_{1}(x)u_{2}(s)}{\alpha (s){\mathcal {W(s)}}}},&a\leq x<s\\{\dfrac {u_{2}(x)u_{1}(s)}{\alpha (s){\mathcal {W}}(s)}},&s<x\leq b\end{cases}}}
where W ( x ) = u 1 ( x ) u 2 ′ ( x ) − u 1 ′ ( x ) u 2 ( x ) {\displaystyle {\mathcal {W}}(x)=u_{1}(x)u'_{2}(x)-u_{1}'(x)u_{2}(x)} is known as the Wronskian determinant of u 1 {\displaystyle u_{1}} and u 2 {\displaystyle u_{2}} . Though this is a somewhat limited case, the Wronskian frequently appears in other sets of boundary value problems that require a one-sided (advanced/retarded) Green's function as well, including those with conditions on boundary derivatives (Neumann conditions) or a pair of conditions on a function and its normal derivative on a single boundary (Cauchy conditions).
Combining Green's functions If the differential operator L {\displaystyle L} can be factored as L = L 1 L 2 {\displaystyle L=L_{1}L_{2}} then the Green's function of L {\displaystyle L} can be constructed from the Green's functions for L 1 {\displaystyle L_{1}} and L 2 {\displaystyle L_{2}} :
G ( x , s ) = ∫ G 2 ( x , s 1 ) G 1 ( s 1 , s ) d s 1 . {\displaystyle G(x,s)=\int G_{2}(x,s_{1})\,G_{1}(s_{1},s)\,ds_{1}.}
The above identity follows immediately from taking G ( x , s ) {\displaystyle G(x,s)} to be the representation of the right operator inverse of L {\displaystyle L} , analogous to how for the invertible linear operator C {\displaystyle C} , defined by C = ( A B ) − 1 = B − 1 A − 1 {\displaystyle C=(AB)^{-1}=B^{-1}A^{-1}} , is represented by its matrix elements C i , j {\displaystyle C_{i,j}} . A further identity follows for differential operators that are scalar polynomials of the derivative, L = P N ( ∂ x ) {\displaystyle L=P_{N}(\partial _{x})} . The fundamental theorem of algebra, combined with the fact that ∂ x {\displaystyle \partial _{x}} commutes with itself, guarantees that the polynomial can be factored, putting L {\displaystyle L} in the form:
L = ∏ i = 1 N ( ∂ x − z i ) , {\displaystyle L=\prod _{i=1}^{N}\left(\partial _{x}-z_{i}\right),}
where z i {\displaystyle z_{i}} are the zeros of P N ( z ) {\displaystyle P_{N}(z)} . Taking the Fourier transform of L G ( x , s ) = δ ( x − s ) {\displaystyle LG(x,s)=\delta (x-s)} with respect to both x {\displaystyle x} and s {\displaystyle s} gives:
G ^ ( k x , k s ) = δ ( k x − k s ) ∏ i = 1 N ( i k x − z i ) . {\displaystyle {\widehat {G}}(k_{x},k_{s})={\frac {\delta (k_{x}-k_{s})}{\prod _{i=1}^{N}(ik_{x}-z_{i})}}.}
The fraction can then be split into a sum using a partial fraction decomposition before Fourier transforming back to x {\displaystyle x} and s {\displaystyle s} space. This process yields identities that relate integrals of Green's functions and sums of the same. For example, if L = ( ∂ x + γ ) ( ∂ x + α ) 2 {\displaystyle L=\left(\partial _{x}+\gamma \right)\left(\partial _{x}+\alpha \right)^{2}} then one form for its Green's function is:
G ( x , s ) = 1 ( γ − α ) 2 Θ ( x − s ) e − γ ( x − s ) − 1 ( γ − α ) 2 Θ ( x − s ) e − α ( x − s ) + 1 γ − α Θ ( x − s ) ( x − s ) e − α ( x − s ) = ∫ Θ ( x − s 1 ) ( x − s 1 ) e − α ( x − s 1 ) Θ ( s 1 − s ) e − γ ( s 1 − s ) d s 1 . {\displaystyle {\begin{aligned}G(x,s)&={\frac {1}{\left(\gamma -\alpha \right)^{2}}}\Theta (x-s)e^{-\gamma (x-s)}-{\frac {1}{\left(\gamma -\alpha \right)^{2}}}\Theta (x-s)e^{-\alpha (x-s)}+{\frac {1}{\gamma -\alpha }}\Theta (x-s)\left(x-s\right)e^{-\alpha (x-s)}\\[1ex]&=\int \Theta (x-s_{1})\left(x-s_{1}\right)e^{-\alpha (x-s_{1})}\Theta (s_{1}-s)e^{-\gamma (s_{1}-s)}\,ds_{1}.\end{aligned}}}
While the example presented is tractable analytically, it illustrates a process that works when the integral is not trivial (for example, when ∇ 2 {\displaystyle \nabla ^{2}} is the operator in the polynomial).
Table of Green's functions
The following table gives an overview of Green's functions of frequently appearing differential operators, where r = x 2 + y 2 + z 2 {\textstyle r={\sqrt {x^{2}+y^{2}+z^{2}}}} , ρ = x 2 + y 2 {\textstyle \rho ={\sqrt {x^{2}+y^{2}}}} , Θ ( t ) {\textstyle \Theta (t)} is the Heaviside step function, J ν ( z ) {\textstyle J_{\nu }(z)} is a Bessel function, I ν ( z ) {\textstyle I_{\nu }(z)} is a modified Bessel function of the first kind, and K ν ( z ) {\textstyle K_{\nu }(z)} is a modified Bessel function of the second kind. Where time (t) appears in the first column, the retarded (causal) Green's function is listed.
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 theorem),
∫ V ∇ ⋅ A d V = ∫ S A ⋅ d σ ^ . {\displaystyle \int _{V}\nabla \cdot \mathbf {A} \,dV=\int _{S}\mathbf {A} \cdot d{\hat {\boldsymbol {\sigma }}}\,.}
Let A = φ ∇ ψ − ψ ∇ φ {\displaystyle \mathbf {A