A Borg–Levinson theorem for trees

Abstract

We prove that the Dirichelet-to-Neumann map for a Schrödinger operator on a finite simply connected tree determines uniquely the potential on that tree.

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].…”

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.…”

Schrödinger Operators on Graphs and Geometry II. Spectral Estimates for $${\varvec{L}}_\mathbf{1}$$ L 1 -potentials and an Ambartsumian Theorem

“…The simplest graph to be considered is the star graph formed by a finite number of compact edges, and the corresponding inverse problem resembles very much the inverse problem on a single interval, where the potential is determined by two spectra [14,30,[33][34][35]. The case of general trees has also been studied and we have a rather good understanding of the problem [2,8,9,31,37]. The case of graphs with cycles is much more involved-major difficulties are related to the reconstruction of the potential on the cycles.…”