On the inverse problem of the two-velocity tree-like graph

TL;DR

This paper proves the unique determination of varying densities and complete structural information of a planar tree-like network of strings using boundary measurements, employing spectral and wave analysis techniques.

Contribution

It introduces a method to recover both densities and the entire graph structure of a tree network from boundary data, advancing inverse problem solutions on metric trees.

Findings

Proves identifiability of densities and graph parameters from boundary data.

Uses Titchmarch-Weyl function and Steklov-Poincaré operator for analysis.

Employs a layer-by-layer peeling approach from leaves to root.

Abstract

In this article the authors continue the discussion in \cite{ALM} about inverse problems for second order elliptic and hyperbolic equations on metric trees from boundary measurements. In the present paper we prove the identifiability of varying densities of a planar tree-like network of strings along with the complete information on the graph, i.e. the lengths of the edges, the edge degrees and the angles between neighbouring edges. The results are achieved using the Titchmarch-Weyl function for the spectral problem and the Steklov-Poincar{\'e} operator for the dynamic wave equation on the tree. The general result is obtained by a peeling argument which reduces the inverse problem layer-by-layer from the leaves to the clamped root of the tree.