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

TL;DR

This paper investigates whether the shape of equilateral quantum trees can be uniquely determined from their spectral data, proving uniqueness for trees with up to 8 vertices and classifying co-spectral trees with 9 vertices.

Contribution

It establishes spectral uniqueness for small equilateral trees and explicitly characterizes all co-spectral trees with 9 vertices.

Findings

No co-spectral trees among equilateral trees with ≤8 vertices.

All co-spectral trees with 9 vertices are identified and described.

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.