  In one dimension one can dissect a scattering potential $ v(x) $ into pieces
$ v_i(x) $ and use the notion of the transfer matrix to determine the
scattering content of $ v(x) $ from that of $ v_i(x) $. This observation has
numerous practical applications in different areas of physics. The problem of
finding an analogous procedure in dimensions larger than one has been an
important open problem for decades. We give a complete solution for this
problem and discuss some of its applications. In particular we derive an exact
expression for the scattering amplitude of the delta-function potential in two
and three dimensions and a potential describing a slab laser with a surface
line defect. We show that the presence of the defect makes the slab begin
lasing for arbitrarily small gain coefficients.
