where
Without loss of generality, we will take λ to be non-negative. When λ is zero, the equation reduces to Poisson's equation. Therefore, when λ is very small, the solution approaches that of the unscreened Poisson equation, which is a superposition of 1/r functions weighted by the source function f:
On the other hand, when λ is extremely large, u approaches the value f/λ², which goes to zero as λ goes to infinity. As we shall see, the solution for intermediate values of λ behaves as a superposition of screened (or damped) 1/r functions, with λ behaving as the strength of the screening.
The screened Poisson equation can be solved for general f using the method of Green's functions[?]. The Green's function G is defined by
Assuming u and its derivatives vanish at large r, we may perform a continuous Fourier transform in spatial coordinates:
where the integral is taken over all space. It is then straightforward to show that
The Green's function in r is therefore given by the inverse Fourier transform,
This integral may be evaluated using spherical coordinates in k-space. The integration over the angular coordinates is straightforward, and the integral reduces to one over the radial coordinate k:
This may be evaluated using contour integration[?]. The result is:
The solution to the full problem is then given by
As we claimed above, this is a superposition of screened 1/r functions, weighted by the source function f and with λ acting as the strength of the screening. The screened 1/r function is often encountered in physics as a screened Coulomb potential[?], also called a "Yukawa potential".
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