2010/04/02 by Colin Guillarmou, Mikko Salo, Leo Tzou · 1 citation
Engineering · Mathematics · Physics and Astronomy · #Combinatorics #Electromagnetic Scattering and Analysis #Energy (signal processing) #Fixed point #Geometry #Inverse #Inverse scattering problem #Mathematical analysis #Mathematical physics #Mathematics #Matrix (chemical analysis) #Numerical methods in engineering #Numerical methods in inverse problems #Operator (biology) #Physics #Quantum mechanics #Riemann surface #Scattering #Surface (topology) #math.AP #math.DG
paper · pdf · doi:10.1007/s00220-011-1224-y
21 pages
arxiv created 2010/04/02 · openalex publication_date 2011/03/22 · arxiv updated 2015/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
On a fixed Riemann surface (M0,g0) with N Euclidean ends and genus g, we show that, under a topological condition, the scattering matrix SV(\la) at frequency \la > 0 for the operator Δ+V determines the potential V if V∈ C1,α(M0)∩ e-γd(⋅,z0)jL^∞(M0) for all γ>0 and for some j∈\1,2\, where d(z,z0) denotes the distance from z to a fixed point z0∈ M0. The topological condition is given by N≥max(2g+1,2) for j=1 and by N≥ g+1 if j=2. In \rr2 this implies that the operator SV(\la) determines any C1,α potential V such that V(z)=O(e-γ|z|2) for all γ>0.