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

Abstract

In this paper, we study the partial data inverse boundary value problem for the Schr ¨odinger 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 H ¨older type in the high frequency regime. To achieve those goals, we used a method by combining the CGO sulution, Runge approximation and Carleman estimate.