Opuscula Math. 44, no. 5 (2024), 689-705
https://doi.org/10.7494/OpMath.2024.44.5.689
Opuscula Mathematica
Recovering the shape of an equilateral quantum tree with the Dirichlet conditions at the pendant vertices
Anastasia Dudko
Oleksandr Lesechko
Vyacheslav Pivovarchik
Abstract. 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.
Keywords: Sturm-Liouville equation, eigenvalue, spectrum, equilateral tree, caterpillar tree, snowflake graph, root, standard conditions, Dirichlet boundary condition, Neumann boundary condition.
Mathematics Subject Classification: 34B45, 34B24, 34L20.
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- Anastasia Dudko
- South Ukrainian National Pedagogical University named after K.D. Ushinsky, Odesa, Ukraine
- Oleksandr Lesechko
- Odesa State Academy of Civil Engineering and Architecture, Odesa, Ukraine
- Vyacheslav Pivovarchik (corresponding author)
https://orcid.org/0000-0002-4649-2333- South Ukrainian National Pedagogical University named after K.D. Ushinsky, Odesa, Ukraine
- Communicated by Alexander Gomilko.
- Received: 2024-01-01.
- Revised: 2024-04-02.
- Accepted: 2024-04-06.
- Published online: 2024-07-01.
