Recovering the shape of a quantum tree by two spectra

TL;DR

This paper demonstrates that the shape of an equilateral quantum tree can be reconstructed from the spectra of Neumann and Dirichlet boundary value problems, with unique determination for snowflake trees.

Contribution

It introduces a method to recover the shape of equilateral quantum trees using spectral data, extending to snowflake trees where the shape is uniquely determined.

Findings

Spectra of Neumann and Dirichlet problems determine the shape of equilateral trees.

Unique shape recovery for snowflake trees from spectral data.

Spectral methods can reconstruct quantum tree geometries.

Abstract

We show how to find the shape of an equilateral tree using the spectra of the Neumann and the Dirichlet problems generated by the Sturm-Liouville equation. In case of snowflake trees the spectra of the Neumann and Dirichlet problems uniquely determine the shape of the tree.