Green's function

In physics and mathematics, Green's function is an auxiliary function in the solution of linear partial differential equations. The function is named for the British mathematician George Green (1793 – 1841)

Let L_x be a given linear differential operator in n variables x = (x₁, x₂, ..., x_n), then the Green function of L_x is the function G(x,y) defined by

$L_{\mathbf {x} }G(\mathbf {x} ,\mathbf {y} )=-\delta (\mathbf {x} -\mathbf {y} ),$

where δ(x-y) is the Dirac delta function. Once G(x,y) is known, any differential equation involving L_x is formally solved,

$L_{\mathbf {x} }\,\phi (\mathbf {x} )=-\rho (\mathbf {x} )\quad \Longrightarrow \quad \phi (\mathbf {x} )=\int \;G(\mathbf {x} ,\mathbf {y} )\;\rho (\mathbf {y} )\;\mathrm {d} \mathbf {y} .$

The proof is by verification,

$L_{\mathbf {x} }\,\phi (\mathbf {x} )=\int \;L_{\mathbf {x} }\;G(\mathbf {x} ,\mathbf {y} )\;\rho (\mathbf {y} )\;\mathrm {d} \mathbf {y} =-\int \;\delta (\mathbf {x} -\mathbf {y} )\;\rho (\mathbf {y} )\mathrm {d} \mathbf {y} =-\rho (\mathbf {x} )$

where in the last step the defining property of the Dirac delta function is used.

The Green function of a linear differential operator may be seen as its inverse, which becomes clear from the matrix equation analogy. Let $\mathbb {L}$ and $\mathbb {G}$ be n×n matrices connected by

$\mathbb {L} \mathbb {G} =\mathbb {I} \quad \Leftrightarrow \quad \left(\mathbb {L} \mathbb {G} \right)_{ij}=-\delta _{ij},$

then the solution of the matrix-vector equation is

$\mathbb {L} {\boldsymbol {\phi }}=-{\boldsymbol {\rho }}\quad \Longrightarrow \quad {\boldsymbol {\phi }}_{i}=\sum _{j}\mathbb {G} _{ij}{\boldsymbol {\rho }}_{j}$

Reference

P. Roman, Advanced Quantum Theory, Addison-Wesley, Reading, Mass. (1965) Appendix 4.