An Inverse Problem for Quantum Trees with Delta-Prime Vertex Conditions

Abstract

In this paper, we consider a non-standard dynamical inverse problem for the wave equation on a metric tree graph. We assume that the so-called delta-prime matching conditions are satisfied at the internal vertices of the graph. Another specific feature of our investigation is that we use only one boundary actuator and one boundary sensor, all other observations being internal. Using the Neumann-to-Dirichlet map (acting from one boundary vertex to one boundary and all internal vertices) we recover the topology and geometry of the graph together with the coefficients of the equations.

An inverse problem for quantum trees with observations at interior vertices

“…[26,27]. We are unaware of any mathematical works treating the inverse problem on general tree graphs with measurements at the internal vertices, except for [8] where the interior vertices are assumed to satisfy delta-prime matching conditions instead of (3), (4). Internal measurements might have advantages in situations where some boundary vertices are inaccessible.…”