Recovering the shape of an equilateral quantum tree with the Dirichlet conditions at the pendant vertices
type: Item
instance of: scholarly article
title: Recovering the shape of an equilateral quantum tree with the Dirichlet conditions at the pendant vertices
author: A. Dudko
author: Олександр Васильович Лесечко
author: V. Pivovarchik
published in: Opuscula Mathematica
publication date: 2024-09-04T00:00:00Z
review text: The paper deals with the boundary value problems for the Sturm-Liouville equations \N\[\N-y_j'' + q_j(x_j) y_j = \lambda y_j, \quad j = 1, 2, \dots, g, \N\]\Non a tree graph (i.e. graph without cycles) with equal edge lengths. The potentials are assumed to be all equal and symmetric. The Dirichlet boundary conditions are imposed in the pendant vertices and the standard matching conditions (continuity and Kirchhoff's ones), in the interior vertices except for the root vertex. The authors consider the recovery of the graph structure from the two spectra, corresponding to the Dirichlet boundary condition and the Neumann one in the root. It is proved that, for caterpillar trees, the two spectra uniquely determine the shape of a graph. However, in general, the uniqueness does not hold. The authors present an example of co-spectral snowflake trees. The technique is based on the structure of characteristic functions and eigenvalue asymptotics.
reviewed by: Natalia Bondarenko
Mathematics Subject Classification ID: 34B45
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34L05
Mathematics Subject Classification ID: 34A55
zbMATH DE Number: 7906392
zbMATH Keywords: Sturm-Liouville equation
zbMATH Keywords: eigenvalue
zbMATH Keywords: spectrum
zbMATH Keywords: equilateral tree
zbMATH Keywords: caterpillar tree
zbMATH Keywords: snowflake graph
zbMATH Keywords: root
zbMATH Keywords: standard conditions
zbMATH Keywords: Dirichlet boundary condition
zbMATH Keywords: Neumann boundary condition
MaRDI profile type: Publication
cites work: Can one hear the shape of a network?
cites work: A characteristic equation associated to an eigenvalue problem on \(c^ 2\)-networks
cites work: Schrödinger operators on graphs and geometry. II: Spectral estimates for \(L_1\)-potentials and an Ambartsumian theorem
cites work: On recovering the shape of a quantum tree from the spectrum of the Dirichlet boundary problem
cites work: Q5445633
cites work: Spectral asymptotics for quantum graphs with equal edge lengths
cites work: Recovering the shape of a quantum graph
cites work: Can one hear the shape of a graph?
cites work: Spectral determinants and an Ambarzumian type theorem on graphs
cites work: Rayleigh estimates for differential operators on graphs
cites work: Sturm-Liouville operators and applications. Transl. from the Russian by A. Iacob
cites work: Direct and Inverse Finite-Dimensional Spectral Problems on Graphs
cites work: Distinguishing cospectral quantum graphs by scattering
cites work: On Ambarzumian type theorems for tree domains
cites work: Recovering the shape of a quantum tree by two spectra
DOI: 10.7494/OPMATH.2024.44.5.689
description: scientific article; zbMATH DE number 7906392