Green's Function--Helmholtz Differential Equation

The inhomogeneous Helmholtz Differential Equation is

$$\nabla^2\psi({\bf r})+k^2\psi({\bf r})=\rho({\bf r}),$$ (1)

where the Helmholtz operator is defined as $\tilde L\equiv \nabla^2+k^2$. The Green's function is then defined by

$$(\nabla^2+k^2)G({\bf r}_1,{\bf r}_2) = \delta^3({\bf r}_1-{\bf r}_2).$$ (2)

Define the basis functions $\phi_n$ as the solutions to the homogeneous Helmholtz Differential Equation

$$\nabla^2\phi_n({\bf r})+{k_n}^2\phi_n({\bf r})=0.$$ (3)

The Green's function can then be expanded in terms of the $\phi_n$s,

$$G({\bf r}_1,{\bf r}_2)=\sum_{n=0}^\infty a_n({\bf r}_2)\phi_n({\bf r}_1),$$ (4)

and the Delta Function as

$$\delta^3({\bf r}_1-{\bf r}_2)=\sum_{n=0}^\infty \phi_n({\bf r}_1)\phi_n({\bf r}_2).$$ (5)

Plugging (4) and (5) into (2) gives

$$\nabla^2 \left[{\sum_{n=0}^\infty a_n({\bf r}_2)\phi_n({\bf ... ...bf r}_1)=\sum_{n=0}^\infty \phi_n({\bf r}_1)\phi_n({\bf r}_2).$$ (6)

Using (3) gives

$$-\sum_{n=0}^\infty a_n({\bf r}_2){k_n}^2\phi_n({\bf r})+k^2\... ...\bf r}_1)=\sum_{n=0}^\infty \phi_n({\bf r}_1)\phi_n({\bf r}_2)$$ (7)

$$\sum_{n=0}^\infty a_n({\bf r}_2)\phi_n({\bf r}_1)(k^2-{k_n}^2) = \sum_{n=0}^\infty \phi_n({\bf r}_1)\phi_n({\bf r}_2).$$ (8)

This equation must hold true for each $n$, so

$$a_n({\bf r}_2)\phi_n({\bf r}_1)(k^2-{k_n}^2) = \phi_n({\bf r}_1)\phi_n({\bf r}_2)$$ (9)

$$a_n({\bf r}_2) = {\phi_n({\bf r}_2)\over k^2-{k_n}^2},$$ (10)

and (4) can be written

$$G({\bf r}_1,{\bf r}_2)=\sum_{n=0}^\infty {\phi_n({\bf r}_1)\phi_n({\bf r}_2)\over k^2-{k_n}^2}.$$ (11)

The general solution to (1) is therefore

$$\psi({\bf r}_1) = \int G({\bf r}_1,{\bf r}_2)\rho({\bf r}_2)\,d^3{\bf r}_2$$

$$= \sum_{n=0}^\infty \int {\phi_n({\bf r}_1)\phi_n({\bf r}_2)\rho({\bf r}_2)\over k^2-{k_n}^2} \, d^3{\bf r}_2.$$ (12)

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 529-530, 1985.