Composition of transfer matrices for potentials with overlapping support

Abstract

For a pair of real or complex scattering potentials v(j) : R -> C (j = 1, 2) with support I-j and transfer matrix M-j, the transfer matrix of v(1) + v(2) is given by the product M2M1 provided that I lies to the left of I-2. We explore the prospects of generalizing this composition rule for the cases that I-1 and I-2 have a small intersection. In particular, we show that if I-1 and I-2 intersect in a finite closed interval of length in which both the potentials are analytic, then the lowest order correction to the above composition rule is proportional to l(5). This correction is of the order of l(3), if v(1) and v(2) are respectively analytic throughout this interval except at x = l and x = 0. We use these results to explore the superposition of a pair of unidirectionally invisible potentials with overlapping support.