Inverse Problem in Cylindrical Electrical Networks

Abstract

In this paper we study the inverse Dirichlet-to-Neumann problem for certain cylindrical electrical networks. We define and study a birational transformation acting on cylindrical electrical networks called the electrical $R$-matrix. We use this transformation to formulate a general conjectural solution to this inverse problem on the cylinder. This conjecture extends work of Curtis, Ingerman, and Morrow [Linear Algebra Appl., 283 (1998), pp. 115--150] and of de Verdière, Gitler, and Vertigan [Comment. Math. Helv., 71 (1996), pp. 144--167] for circular planar electrical networks. We show that our conjectural solution holds for certain „purely cylindrical” networks. Here we apply the grove combinatorics introduced by Kenyon and Wilson [Trans. Amer. Math. Soc., 363 (2011), pp. 1325--1364].