Inverse scattering at fixed energy on surfaces with Euclidean ends

Details Inverse scattering at fixed energy on surfaces with Euclidean ends Inverse scattering at fixed energy on surfaces with Euclidean ends

2010-04-02

arxiv.org/abs/1004.0315v1

Mathematics

Analysis of PDEs

21 pages

Scientific paper

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 $\gamma>0$.

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