Stability estimates for an inverse problem for schrodinger operators at high frequencies from arbitrary partial boundary measurements

In this paper, we study the partial data inverse boundary value problem for the Schrodinger operator at a high frequency k>=1 in a bounded domain with smooth boundary in Rn, n>=3. Assuming that the potential is known in a neighborhood of the boundary, we obtain the logarithmic stability when both Dirichlet data and Neumann data are taken on arbitrary open subsets of the boundary where the two sets can be disjointed. Our results also show that the logarithmic stability can be improved to the one of Holder type in the high frequency regime. To achieve those goals, we used a method by combining the CGO sulution, Runge approximation and Carleman estimate.