Integral equation approach to the problem of scattering by a number of fixed scatterers

In the derivation of the Lippmann–Schwinger integral equation for scattering of a wave ψ(r) by the potential ν(r), one constructs the Green's function for the operator [Formula: see text], and treats νψ as the inhomogeneous term. However, in certain cases, it is desirable to formulate the scattering problem in terms of an integral equation by obtaining the Green's function for the operator [Formula: see text], and by considering (−k2ψ) as the inhomogeneous term. An important aspect of this formulation is that the resulting integral equation can be used to generate a low energy expansion of the wave function for some separable and nonseparable systems. For two-dimensional scattering, if the geometry of the scatterers is simple enough, the Laplace equation with the prescribed boundary conditions on the surface of the scatterers is separable in a certain coordinate system, then one can write the solution of the wave equation as an inhomogeneous integral equation. In this way the problems of scattering by two cylinders, an array of cylinders, and a grating can be formulated in terms of integral equations. For three-dimensional scattering, one can consider either the spherically symmetric cases or nonseparable problems. In the former case, for certain types of force laws, a Volterra integral equation in one variable can be found for the wave function. In the latter case, integral equations in two or three variables can be obtained for scattering by two spheres or by a torus.