Fernando Guevera Vasquez, fguevara@math.utah.edu
Co-authors: Liliana Borcea, Alexander V. Mamonov
A network based inversion method for the Schrödinger problem
Consider the inverse problem of finding the Schrödinger potential q ( x ) from measurements of the Dirichlet to
Neumann map of the Schrödinger differential operator u → ∆ u + qu . We present a method for solving this problem in 2D that is based on an analogous of the Liouville transform on resistor networks. The continuum
Liouville transform maps an operator of the form v → ∇ · [ σ ∇ v ] , with σ > 0 to a Schrödinger differential operator. The discrete Liouville transform on a network maps a Laplacian on the network to another Laplacian plus a diagonal perturbation which can be thought as a discrete Schrödinger potential. The first step in the inversion is to find a resistor network that fits the data. In the second step, a discrete Liouville transform is applied to get a diagonal perturbation of a discrete Laplacian. The final step is to interpret the discrete Laplacian and its diagonal perturbation as a finite volumes discretization of the Schrödinger differential operator. This gives a cheap reconstruction of the continuum Schrödinger potential from the discrete one. The method needs to be calibrated by data for two known constant Schrödinger potentials.