Boundary spectral inverse problem on a class of graphs (trees) by the BC method

Abstract

A planar graph consisting of strings of variable densities is considered. The spectrum of the Dirichlet problem on the graph and the values of derivatives of the (normalized) eigenfunctions at the boundary vertices form the spectral data. We show that the graph without cycles (tree) and the densities of its edges are determined by the spectral data uniquely up to a natural isometry in the plane. In the framework of our approach (boundary control method; Belishev 1986) we study the boundary controllability of the dynamical system associated with the graph and governed by the wave equation, and exploit this property for recovering the tree from its spectral data. The approach can be extended to a wide class of inverse problems on trees.

Inverse problems on star-type graphs: differential operators of different orders on different edges

“…For second-order differential operators on compact graphs inverse spectral problems have been studied fairly completely in [3,4,25,28,[31][32][33] and other works. Inverse problems for higher-order differential operators on graphs were investigated in [29,30].…”

An adjoint method for channel localization

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Inverse scattering problem for Sturm-Liouville operator on one-vertex noncompact graph with a cycle

“…During the last years such problems were in the focus of intensive investigations. The most complete results on (both direct and inverse) spectral problems were achieved in the case of compact graphs [8], [7], [9], [10], [11], [12], [13]. In the noncompact case there are no similar general results since the presence of the noncompact edges (rays) leads to new qualitative difficulties in the investigation of the spectral problems.…”