On a fixed smooth compact Riemann surface with boundary (M 0 , g), we show that, for the Schrödinger operator g + V with potential V ∈ C 1,α (M 0 ) for some α > 0, the Dirichlet-to-Neumann map N | measured on an open set ⊂ ∂M 0 determines uniquely the potential V . We also discuss briefly the corresponding consequences for potential scattering at zero frequency on Riemann surfaces with either asymptotically Euclidean or asymptotically hyperbolic ends.

We consider limiting Carleman weights for Dirac operators and prove corresponding Carleman estimates. In particular, we show that limiting Carleman weights for the Laplacian also serve as limiting weights for Dirac operators. As an application we consider the inverse problem of recovering a Lipschitz continuous magnetic field and electric potential from boundary measurements for the Pauli Dirac operator.

In this paper we establish a log log-type estimate which shows that in dimension n ≥ 3 the magnetic field and the electric potential of the magnetic Schrödinger equation depends stably on the Dirichlet to Neumann (DN) map even when the boundary measurement is taken only on a subset that is slightly larger than half of the boundary ∂Ω. Furthermore, we prove that in the case when the measurement is taken on all of ∂Ω one can establish a better estimate that is of log-type. The proofs involve the use of the complex geometric optics (CGO) solutions of the magnetic Schrödinger equation constructed in [8] then follow a similar line of argument as in [1]. In the partial data estimate we follow the general strategy of [5] by using the Carleman estimate established in [4] and a continuous dependence result for analytic continuation developed in [14].

We consider a connection ∇ X on a complex line bundle over a Riemann surface with boundary M 0 , with connection 1-form X. We show that the Cauchy data space of the connection Laplacian (also called magnetic Laplacian) L := ∇ X * ∇ X + q, with q a complex-valued potential, uniquely determines the connection up to gauge isomorphism, and the potential q.