The spectrum of Dirichlet-to-Neumann maps for radial conductivities

Abstract

The problem of characterizing sequences of real numbers that arise as spectra of Dirichlet-to-Neumann (DtN) maps for elliptic operators has attracted considerable attention over the past fifty years. In this article, we address this question in the simple setting of DtN maps associated with a rotation-invariant elliptic operator $\nabla \cdot (\gamma\nabla \centerdot )$ in the ball in Euclidean space. We show that the spectrum of such a DtN operator can be expressed as a universal term, determined solely by the boundary values of the conductivity $\gamma$, plus a sequence of Hausdorff moments of an integrable function, which we call the Born approximation of $\gamma$. We also show that this object is locally determined from the boundary by the corresponding values of the conductivity, a property that implies a local uniqueness result for the Calder\'on Problem in this setting. We also give a stability result: the functional mapping the Born approximation to its conductivity is H\"older stable in suitable Sobolev spaces. Finally, in order to refine the characterization of the Born approximation, we analyze its regularity properties and their dependence on the conductivity.