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Recovering the shape of an equilateral quantum tree with the Dirichlet conditions at the pendant vertices

Dudko Anastasia, Lesechko Oleksandr, Pivovarchik Vyacheslav

Opuscula Mathematica

2024

Vol. 44, no. 5

689--705

We consider two spectral problems on an equilateral rooted tree with the standard (continuity and Kirchhoff’s type) conditions at the interior vertices (except of the root if it is interior) and Dirichlet conditions at the pendant vertices (except of the root if it is pendant). For the first (Neumann) problem we impose the standard conditions (if the root is an interior vertex) or Neumann condition (if the root is a pendant vertex) at the root, while for the second (Dirichlet) problem we impose the Dirichlet condition at the root. We show that for caterpillar trees the spectra of the Neumann problem and of the Dirichlet problem uniquely determine the shape of the tree. Also, we present an example of co-spectral snowflake graphs.

On Ambarzumian type theorems for tree domains

Pivovarchik Vyacheslav

Opuscula Mathematica

2022

Vol. 42, no. 3

427--437

It is known that the spectrum of the spectral Sturm–Liouville problem on an equilateral tree with (generalized) Neumann’s conditions at all vertices uniquely determines the potentials on the edges in the unperturbed case, i.e. case of the zero potentials on the edges (Ambarzumian’s theorem). This case is exceptional, and in general case (when the Dirichlet conditions are imposed at some of the pendant vertices) even two spectra of spectral problems do not determine uniquely the potentials on the edges. We consider the spectral Sturm–Liouville problem on an equilateral tree rooted at its pendant vertex with (generalized) Neumann conditions at all vertices except of the root and the Dirichlet condition at the root. In this case Ambarzumian’s theorem can’t be applied. We show that if the spectrum of this problem is unperturbed, the spectrum of the Neumann-Dirichlet problem on the root edge is also unperturbed and the spectra of the problems on the complimentary subtrees with (generalized) Neumann conditions at all vertices except the subtrees’ roots and the Dirichlet condition at the subtrees’ roots are unperturbed then the potential on each edge of the tree is 0 almost everywhere.