Composition of Transfer Matrices for Potentials with Overlapping Support

Abstract

For a pair of real or complex scattering potentials $v_j:\mathbb{R}\to\mathbb{C}$ ($j=1,2$) with support $I_j$ and transfer matrix $M_j$, the transfer matrix of $v_1+v_2$ is given by the product $M_2 M_1$ provided that $I_1$ 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 $\ell$ in which both the potentials are analytic, then the lowest order correction to the above composition rule is proportional to $\ell^5$. This correction is of the order of $\ell^3$, if $v_1$ and $v_2$ are respectively analytic throughout this interval except at $x=\ell$ and $x=0$. We use these results to explore the superposition of a pair of unidirectionally invisible potentials with overlapping support.