# Uniqueness in inverse acoustic scattering with unbounded gradient across Lipschitz surfaces

## TL;DR

This paper establishes uniqueness in inverse acoustic scattering problems involving media with unbounded gradient across Lipschitz surfaces, by linking it to Schrödinger operators with singular delta potentials.

## Contribution

It introduces a novel uniqueness result for inverse scattering with media having unbounded gradient, extending previous theories to include singular surface-supported potentials.

## Key findings

- Proves uniqueness in inverse acoustic scattering with unbounded gradient.
- Establishes a connection between acoustic scattering and Schrödinger operators with delta potentials.
- Extends inverse scattering theory to more singular media configurations.

## Abstract

We prove uniqueness in inverse acoustic scattering in the case the density of the medium has an unbounded gradient across $\Sigma\subseteq\Gamma=\partial\Omega$, where $\Omega$ is a bounded open subset of $\mathbb{R}^{3}$ with a Lipschitz boundary. This follows from a uniqueness result in inverse scattering for Schr\"odinger operators with singular $\delta$-type potential supported on the surface $\Gamma$ and of strength $\alpha\in L^{p}(\Gamma)$, $p>2$.