One-dimensional Scattering Problem: Exterior Boundary Value Problem

Scattering of radiation by bodies is ubiquitous. The mathematical formulation of this process can be reduced, more often than not, to an exterior boundary value problem, namely finding the solution to the following homogeneous b.v. problem:

$$-\frac{d}{dx}p(x)\frac{d\psi}{dx}-q(x)\psi =\lambda\rho(x)\psi\quad a<x<\infty$$ (443)

$$\psi(a) =0$$ (444)

$$\lim_{x\to\infty}\psi(x) =finite$$ (445)

The scattering is due to $q(x)$ , the potential of the body. The body is finite. Consequently, its potential vanishes for large $x$ :

$$\lim_{x\to\infty} q(x)=0~.$$

If the scattering body is absent, there is no scattering potential at all. In that case the boundary value problem is

$$-\frac{d}{dx}p(x)\frac{d\psi_0}{dx} =\lambda\rho(x)\psi_0\quad a<x<\infty$$ (446)

$$\psi_0(a) =0$$ (447)

$$\lim_{x\to\infty}\psi_0(x) =finite ~.$$ (448)

The solution to this problem, $\psi_0$ , is called the unscattered solution or the incident wave. It is characterized by the physical parameter $\lambda$ , which usually expresses a squared frequency, a squared wave number, an energy or something else, depending on the nature of the wave.

If the scattering body is present, $q(x)\ne 0$ . To find the scattered wave, i.e. the solution to the homogeneous Eqs.(4.43)-(4.45), one writes this system with the help of Eqs.(4.46)-(4.48) in the form

$$\frac{d}{dx}p(x)\frac{d\{\psi(x)-\psi_0(x)\}}{dx} +\lambda\rho(x)\{\psi(x)-\psi_0(x)\} =-q(x)\psi\quad a<x<\infty$$ (449)

$$\{ \psi(a)-\psi_0(a)\} =0$$ (450)

$$\lim_{x\to\infty}\{\psi(x)-\psi_0(x)\} =0~,$$ (451)

and views the r.h.s., $q(x)\psi(x)$ , as an inhomogeneity for the corresponding Green's function problem

$$\left[\frac{d}{dx}p(x)\frac{d\psi}{dx}+\lambda\rho(x)\right]G_\lambda(x;\xi) =-\delta(x-\xi)\quad a<x<\infty$$

$$G_\lambda(a;\xi) =0$$

$$\lim_{x\to\infty}G_\lambda(x;\xi) =0~.$$

Solutions to problems like this one are discussed in the next Section 4.10.3 starting on page . It follows from Eq.(4.15) on page  that

$$\psi(x)=\psi_0(x)+\int_a^\infty G_\lambda(x;x')q(x')\psi(x')\,dx'~.$$ (452)

However, unlike Eq.(4.15), Eq.(4.52) is not an explicit solution because the unknown function $\psi$ appears inside the integral. The physical reason is that the source of the scattered wave on the r.h.s. of Eq.(4.49) is nonzero if and only if an incident wave $\psi_0$ is present, i.e.

$$\psi_0(x)\ne 0 \Longleftrightarrow \psi(x)\ne 0 \quad a<x<\infty~.$$

Equation (4.52) is an integral equation for the to-be-determined solution $\psi(x)$ . This equation not only implies the differential Eq.(4.43), but also the associated boundary conditions. In fact, the integral equation is mathematically equivalent to the homogeneous boundary value problem, Eqs.(4.43)-(4.45).

The reformulation in terms of an integral equation constitutes a step forward. By condensing three concepts into one, one has implemented the principle of unit-economy⁴¹, and thereby identified the essence - the most consequential aspect - of the external boundary value problem. That this is indeed the case is borne out by the fact that Eq.(4.52) lends itself to being solved by a process of iteration without having to worry about boundary conditions.

The first iterative term, with $\psi(x')$ inside the integral replaced by $\psi_0(x')$ , yields what in scattering theory is called the first Born approximation:

$$\psi_1(x) =\psi_0(x)+\int_a^\infty G_\lambda(x;x')q(x')\psi_0(x')\,dx'$$

$$=\psi_0(x)+\Delta^{(1)}\psi(x)~.$$

Successive terms in this iteration yield

$$\psi_2(x) =\psi_0(x)+\int_a^\infty G_\lambda(x;x')q(x')\psi_1(x')\,dx'$$

$$=\overbrace{ \psi_0(x)+\int_a^\infty G_\lambda(x;x')q(x')\psi_0(x')\,dx' }^{\psi_1(x)}$$

$$~~~~~~~~~~~~~~ +\int_a^\infty \int_a^\infty G_\lambda(x;x')q(x')G_\lambda(x';x'')q(x'') \psi_0(x'')\,dx''dx'$$

$$\equiv \psi_0(x)+\Delta^{(1)}\psi(x)+\Delta^{(2)}\psi(x)$$

$$~\vdots$$

The $n^{\textrm{th}}$ iteration involves a multiple integral of the form

$$\Delta^{(n)}\psi(x)= \int_a^\infty \int_a^\infty \cdots\int_a^\infty G_\lambda(x;x')q(x') G_\lambda(x';x'')q(x'')$$

$$\cdots G_\lambda(x^{(n-1)};x^{(n)})q(x') \psi_0(x'')\,dx^{(n)}\cdots dx''dx'$$

For $n=1$ such a term corresponds to a scattering process in which the incident wave is scattered by the potential at $x'$ before it arrives at $x$ . The integration over $x'$ expresses the fact that the total total wave amplitude at $x$ is a linear superposition of the waves due to the scattering process taking place at $x'$ .

By induction one concludes that for any $n$ such a term refers to a multiple scattering process: the incident wave is scattered by the potential at $x^{(n)},x^{(n-1)},\cdots,x'',x'$ before it arrives at $x$ , where it is observed.

Thus the solution to the external boundary value problem, Eq.(4.52), has the form

$$\psi(x)= \psi_0(x)+\Delta^{(1)}\psi(x)+\Delta^{(2)}\psi(x)+\cdots+ \Delta^{(n)}\psi(x)+\cdots~.$$

The scattered wave is represented by a Born series, a sum of the unscattered wave $\psi_0(x)$ , a wave $\Delta^{(1)}\psi(x)$ that was scattered once, a wave $\Delta^{(2)}\psi(x)$ that was scattered twice, and so on. The Born series converges if the scattering potential is small enough. By truncating this series one obtains an approximate solution to the given exterior boundary value problem.

Footnotes

... unit-economy⁴¹

As identified in the footnote on page