Composition of Transfer Matrices for Potentials with Overlapping Support

TL;DR

This paper investigates how transfer matrices for overlapping potentials can be approximately composed, revealing that the correction depends on the overlap length and analyticity, with implications for designing unidirectionally invisible potentials.

Contribution

It generalizes the composition rule for transfer matrices to overlapping potentials, quantifying the correction terms based on overlap size and analyticity conditions.

Findings

Correction to transfer matrix composition is proportional to overlap length to the fifth power.

Analyticity of potentials reduces correction order to cubic in overlap length.

Results enable analysis of superposition of unidirectionally invisible 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.