Local uniqueness for the Dirichlet-to-Neumann map via the two-plane transform

We consider the Dirichlet-to-Neumann map associated to the Schrö– dinger equation with a potential on a bounded domain Ω ⊂ Rn, n ≥ 3. We show that the integral of the potential over a two-plane Π is determined by the values of the integral kernel of the Dirichlet-to-Neumann map on any open subset U ⊂ ∂Ω which contains Π ∩ ∂Ω. 0 Introduction For Ω a bounded domain in R with Lipschitz boundary, ∂Ω, and q(x) ∈ L∞(Ω) , let Λq : H 1 2 (∂Ω) → H− 1 2 (∂Ω) (0.1) be the Dirichlet-to-Neumann map associated with the operator ∆ + q on Ω. (We assume throughout that λ = 0 is not a Dirichlet eigenvalue for ∆ + q on Ω). If Ω and q(x) are C∞, then Λq is a first order ΨDO, with an integral kernel Kq: Λqf(x) = ∫ ∂Ω Kq(x, y)f(y)dσ(y), x ∈ ∂Ω. (0.2) Partially supported by an NSF grant. Partially supported by an NSF grant and the Royal Research Fund at the University of Washington