Composition of transfer matrices for potentials with overlapping support

For a pair of real or complex scattering potentials vj:R→C (j=1,2) with support Ij and transfer matrix Mj, the transfer matrix of v1+v2 is given by the product M2M1 provided that I1 lies to the left of I2. We explore the prospects of generalizing this composition rule for the cases that I1 and I2 have a small intersection. In particular, we show that if I1 and I2 intersect in a finite closed interval of length ℓ in which both the potentials are analytic, then the lowest order correction to the above composition rule is proportional to ℓ5. This correction is of the order of ℓ3, if v1 and v2 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.