Inverse scattering at fixed energy on surfaces with Euclidean ends

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 $N\ge \max (2g+1,2)$ for $j=1$ and by $N\ge g+1$ 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 $\gamma >0$.

21 pages

References in corpus (3)

• Calderon inverse Problem with partial data on Riemann Surfaces
• Global uniqueness from partial Cauchy data in two dimensions
• Inverse scattering for the magnetic Schroedinger operator

Cited by in corpus (4)

• On the Gel'fand-Calderón inverse problem in two dimensions
• Linearization stability results and active measurements for the Einstein-scalar field equations
• Inverse scattering on non-compact manifolds with general metric
• Unique determination of a magnetic Schrödinger operator with unbounded magnetic potential from boundary data