what is scattering theory?

Question

I often read the the words "scattering theory", "scattering data", "scattering matrix", scattering XXX ... in my math lecture, but I realised that I am not able to define it correctly. A short search on google, did not provide me a satisfactory mathematical answer? Can anyone give me a precise définition with some concrete example of what is this the mathematical concept behind all this "scattering" ?

Answer 1

I will consider the case of wave scattering in three dimensions. Scattering theory concerns the behavior of a wave $u(\mathbf{x})$ impacted by some obstacle, e.g., an aperture, an object, etc., in some region $\mathcal{R} \subset \mathbb{R}^3$. We may assume here that the wave satisfies the Helmholtz equation

$$\nabla^2 u + k^2 u = 0$$

There are boundary conditions dictated by the geometry and physical properties of the obstacle (e.g., index of refraction). In the case of a plane aperture, these are relatively straightforward. In the case of a 3D object that is partially transparent, these may be more complex.

More interesting, however, is the behavior of scattered waves at infinity. Here, an analog of the Sommerfeld radiation (boundary) condition at infinity is a statement of the asymptotic behavior of the wave $u(\mathbf{x})$ at infinity:

$$u(\mathbf{x}) \sim f(\theta,\phi) \frac{e^{i k |\mathbf{x}|}}{|\mathbf{x}|} \quad (|\mathbf{x}| \to \infty)$$

Scattering theory is concerned with determining $f$. $f$ is more interesting from a physics perspective because it is what is typically measurable. In many cases, one may even determine the composition of the scattering object from $f$. (This is inverse scattering theory and is mathematically fascinating in its own right.)

A concrete example of this is what is known as Fraunhofer diffraction of waves by an aperture in a plane. In this case, it turns out that $f$ is related to the Fourier transform of the function defined by the aperture.

Answer 2

Let $V:\mathbb R\to\mathbb R$ be a smooth function such that $V(x)=0$ if $x<-1$ or if $x>1$. Consider on one hand the differential equation $$u''(x)+u(x)=V(x)$$ and, on the other, $$u''(x)+u(x)=0.$$ The vector space $V$ of solutions of the second one is $2$-dimensional, with $\cos x$ and $\sin x$ as basis. The vector space $W$ of solutions of the first equation is also $2$-dimensional, but we do not really know exactly what the solutions are.

Suppose $f\in V$ is a solution of the second equation. There is exactly one solution $g\in W$ of the first equation such that $f(x)=g(x)$ for all $x<-1$. Also, there is a unique solution $h\in V$ of the second equation such that $h(x)=g(x)$ for all $x>1$. The rule mapping $f$ to $h$ is well-defined, and gives us a linear map $S:V\to V$. All this is easily proved, using the baasic theory of linear differential equations: consider it an exercise :-)

Scattering theory is the art of deducing information about the function $V$ from information about the linear map $S$, like its eigenvalues and so on.