Inverse scattering at fixed energy on surfaces with Euclidean ends

TL;DR

This paper proves that the scattering matrix at fixed energy uniquely determines certain decaying potentials on Riemann surfaces with Euclidean ends, extending inverse scattering results to more general geometric settings.

Contribution

It establishes new inverse scattering results on Riemann surfaces with Euclidean ends, identifying conditions under which the potential is uniquely recoverable from the scattering matrix.

Findings

The scattering matrix determines potentials with super-exponential decay.

Topological conditions relate the number of ends to the genus for uniqueness.

Results apply to potentials in $C^{1,eta}$ with specific decay rates.

Abstract

On a fixed Riemann surface $(M_0,g_0)$ with $N$ Euclidean ends and genus $g$, we show that, under a topological condition, the scattering matrix $S_V(\la)$ at frequency $\la > 0$ for the operator $\Delta+V$ determines the potential $V$ if $V\in C^{1,\alpha}(M_0)\cap e^{-\gamma d(\cdot,z_0)^j}L^\infty(M_0)$ for all $\gamma>0$ and for some $j\in\{1,2\}$, where $d(z,z_0)$ denotes the distance from $z$ to a fixed point $z_0\in M_0$. The topological condition is given by $N\geq\max(2g+1,2)$ for $j=1$ and by $N\geq g+1$ if $j=2$. In $\rr^2$ this implies that the operator $S_V(\la)$ determines any $C^{1,\alpha}$ potential $V$ such that $V(z)=O(e^{-\gamma|z|^2})$ for all $\gamma>0$.