Green's Function - Definition and Uses

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 Rn, at a point s, is any solution of

(1)

where is the Dirac delta function. This property of a Green's function can be exploited to solve differential equations of the form

(2)

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. Also, Green's functions in general are distributions, not necessarily proper functions.

Green's functions are also a useful tool in solving wave equations, diffusion equations, 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. 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.

If the operator is translation invariant, that is when L has constant coefficients with respect to x, then the Green's function can be taken to be a convolution operator, that is,

In this case, the Green's function is the same as the impulse response of linear time-invariant system theory.