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On recovering the shape of a quantum tree from the spectrum of the Dirichlet boundary problem

Olga Boyko, Olga Martynyuk, Vyacheslav Pivovarchik

Abstract

Spectral problems are considered generated by the Sturm-Liouville equation on equilateral trees with the Dirichlet boundary conditions at the pendant vertices and continuity and Kirchhoff's conditions at the interior vertices. It is proved that there are no co-spectral (i.e., having the same spectrum of such problem) among equilateral trees of less or equal 8 vertices. All co-spectral trees of 9 vertices are presented.

On recovering the shape of a quantum tree from the spectrum of the Dirichlet boundary problem

Abstract

Spectral problems are considered generated by the Sturm-Liouville equation on equilateral trees with the Dirichlet boundary conditions at the pendant vertices and continuity and Kirchhoff's conditions at the interior vertices. It is proved that there are no co-spectral (i.e., having the same spectrum of such problem) among equilateral trees of less or equal 8 vertices. All co-spectral trees of 9 vertices are presented.
Paper Structure (5 sections, 5 theorems, 52 equations, 16 figures)

This paper contains 5 sections, 5 theorems, 52 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Statement of the problems
  3. Auxiliary results
  4. Inverse problem
  5. Co-spectral non-isomorphic equilateral trees of 9 vertices

Key Result

Theorem 1

Let $G$ be a simple connected graph with at least two edges. Assume that all edges have the same length $l$ and the same potentials symmetric with respect to the midpoints of the edges ($q(l-x)=q(x)$). Then the spectrum of problem (2.1)--(2.5) coincides with the set of zeros of the characteristic fu where $s(\sqrt{\lambda},x)$ and $c(\sqrt{\lambda},x)$ are the solutions of the Sturm-Liouville equa

Figures (16)

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  • ...and 11 more figures

Theorems & Definitions (9)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • proof : Proof
  • Remark 1
  • Remark 2
  • Theorem 3
  • Theorem 4
  • Remark 3