Recovering the shape of a quantum caterpillar tree by two spectra

TL;DR

This paper demonstrates that the shape of an equilateral caterpillar tree can be uniquely reconstructed using spectral data from Neumann and Dirichlet boundary value problems associated with a Sturm-Liouville equation on the tree.

Contribution

It establishes a unique determination of the tree's shape from spectral data, advancing inverse spectral theory on quantum graphs.

Findings

Spectra of Neumann and Dirichlet problems uniquely determine the tree shape.

Proof of uniqueness for shape reconstruction in equilateral caterpillar trees.

Method applicable to inverse problems on quantum graphs.

Abstract

We show how to find the shape of an equilateral caterpillar tree using the spectra of the Neumann and the Dirichlet problems generated by the Sturm-Liouville equation on this tree. We prove that in the case of a caterpillar tree the spectra of the Neumann and Dirichlet problems uniquely determine the shape of the tree.