# Quasi-Exactly Solvable Scattering Problems, Exactness of the Born Approximation, and Broadband Unidirectional Invisibility in Two Dimensions

## TL;DR

This paper introduces a method to achieve broadband unidirectional invisibility in two-dimensional scattering problems by extending quasi-exact solvability, enabling the Born approximation to be exact within a specific frequency range.

## Contribution

It extends quasi-exact solvability to scattering theory and provides a simple condition for the Born approximation to be exact, leading to new classes of potentials with desired scattering properties.

## Key findings

- Exact unidirectional invisibility achieved in a finite frequency band.
- Identified classes of potentials with specific scattering features.
- Optical realization using isotropic active media that are non-scattering in certain directions.

## Abstract

Achieving exact unidirectional invisibility in a finite frequency band has been an outstanding problem for many years. We offer a simple solution to this problem in two dimensions that is based on our solution to another more basic open problem of scattering theory, namely finding scattering potentials $v(x,y)$ in two dimensions whose scattering problem is exactly solvable for energies not exceeding a critical value $E_c$. This extends the notion of quasi-exact solvability to scattering theory and yields a simple condition under which the first Born approximation gives the exact expression for the scattering amplitude whenever the wavenumber for the incident wave is not greater than $\alpha:=\sqrt{E_c}$. Because this condition only restricts the $y$-dependence of $v(x,y)$, we can use it to determine classes of such potentials that have certain desirable scattering features. This leads to a partial inverse scattering scheme that we employ to achieve perfect broadband unidirectional invisibility in two dimensions. We discuss an optical realization of the latter by identifying a class of two-dimensional isotropic active media that do not scatter incident TE waves with wavenumber in the range $(\alpha/\sqrt 2,\alpha]$ and source located at $x=\infty$, while scattering the same waves if their source is relocated to $x=-\infty$.