Screened Poisson equation

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In mathematics, the screened Poisson equation is the following partial differential equation:

$\left[ \nabla^2 - \lambda^2 \right] u(\mathbf{r}) = - f(\mathbf{r})$

where ² is the Laplace operator, λ is a constant, f is an arbitrary function of position (known as the "source function") and u is the function to be determined. The screened Poisson equation occurs frequently in physics, including Yukawa's theory of mesons and electric field screening in plasmas.

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:

$u(\mathbf{r})_{(Poisson)} = \int d^3r' \frac{f(\mathbf{r}')}{4\pi |\mathbf{r} - \mathbf{r}'|}.$

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

$\left[ \nabla^2 - \lambda^2 \right] G(\mathbf{r}) = - \delta^3(\mathbf{r}).$

Assuming u and its derivatives vanish at large r, we may perform a continuous Fourier transform in spatial coordinates:

$G(\mathbf{k}) = \int d^3r \; G(\mathbf{r}) e^{i \mathbf{k} \cdot \mathbf{r}}$

where the integral is taken over all space. It is then straightforward to show that

$\left[ k^2 + \lambda^2 \right] G(\mathbf{k}) = 1.$

The Green's function in r is therefore given by the inverse Fourier transform,

$G(\mathbf{r}) = \frac{1}{(2\pi)^3} \; \int d^3\!k \; \frac{e^{-i \mathbf{k} \cdot \mathbf{r}}}{k^2 + \lambda^2}.$

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:

$G(\mathbf{r}) = \frac{1}{2\pi^2 r} \; \int_0^{\infty} dk \; \frac{k \, \sin kr }{k^2 + \lambda^2}.$

This may be evaluated using contour integration. The result is:

$G(\mathbf{r}) = \frac{e^{- \lambda r}}{4\pi r}.$

The solution to the full problem is then given by

$u(\mathbf{r}) = \int d^3r' G(\mathbf{r} - \mathbf{r}') f(\mathbf{r}') = \int d^3r' \frac{e^{- \lambda |\mathbf{r} - \mathbf{r}'|}}{4\pi |\mathbf{r} - \mathbf{r}'|} f(\mathbf{r}').$

As stated 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".