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

• Autorzy: Dudko Anastasia , Lesechko Oleksandr , Pivovarchik Vyacheslav

Identyfikatory

DOI

10.7494/OpMath.2024.44.5.689

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Sturm-Liouville equation; eigenvalue; spectrum; equilateral tree; caterpillar tree; snowflake graph; root; standard conditions; Dirichlet boundary condition; Neumann boundary condition

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2024
• Tom: Vol. 44, no. 5
• Strony: 689--705
• Opis fizyczny: Bibliogr. 20 poz., rys.

Twórcy

autor

Dudko Anastasia

• South Ukrainian National Pedagogical University named after K.D. Ushinsky, Odesa, Ukraine

autor

Lesechko Oleksandr

• Odesa State Academy of Civil Engineering and Architecture, Odesa, Ukraine

autor

Pivovarchik Vyacheslav

• South Ukrainian National Pedagogical University named after K.D. Ushinsky, Odesa, Ukraine

• https://orcid.org/0000-0002-4649-2333

Bibliografia

• [1] J. von Below, Can one hear the shape of a network?, [in:] Partial Differential Equations on Multistructures (Proc. Luminy 1999), Lect. Notes Pure Appl. Math., vol. 219, New York, 2001, 19–36.
• [2] J. von Below, A characteristic equation associated with an eigenvalue problem on c2-networks, Linear Algebra Appl. 71 (1985), 309–325.
• [3] J. Boman, P. Kurasov, R. Suhr, Schrödinger operators on graphs and geometry II. Spectral estimates for L1-potentials and Ambartsumian’s theorem, Integral Equations Operator Theory 90 (2018), Article no. 40.
• [4] G. Borg, Uniqueness theorems in the spectral theory of y′′ + (λ − q(x))y = 0, Proc. 11th Scandinavian Congress of Mathematicians, Johan Grundt Tanums Forlag Oslo, (1952), 00020–01.
• [5] O. Boyko, O. Martynyuk, V. Pivovarchik, On recovering the shape of a quantum tree from the spectrum of the Dirichlet boundary problem, Matematychni Studii 60 (2023), no. 2.
• [6] D.M. Cvetković, M. Doob, H. Sachs, Spectra of Graphs – Theory and Applications, Pure Appl. Math. Academic Press, New York, 1979.
• [7] R. Carlson, V. Pivovarchik, Ambarzumian’s theorem for trees, Electron. J. Differential Equations 2007 (2007), no. 142, 1–9.
• [8] R. Carlson, V. Pivovarchik, Spectral asymptotics for quantum graphs with equal edge lengths, J. Phys. A: Math. Theor. 41 (2008) 145202, 16 pp.
• [9] A. Chernyshenko, V. Pivovarchik, Recovering the shape of a quantum graph, Integral Equations Operator Theory 92 (2020), Article no. 23.
• [10] B. Gutkin, U. Smilansky, Can one hear the shape of a graph?, J. Phys. A: Math. Gen. 34 (2001), 6061–6068.
• [11] D. Kaliuzhnyi-Verbovetskyi, V. Pivovarhik, Recovering the shape of quantum caterpillar graph by two spectra, Mechanics and Mathematical Models 5 (2023), no. 1, 14–24.
• [12] M. Kiss, Spectral determinants and an Ambarzumian type theorem on graphs, Integral Equations Operator Theory 92 (2020), Article no. 24.
• [13] P. Kurasov, S. Naboko, Rayleigh estimates for differential operators on graphs, J. Spectr. Theory 4 (2014), no. 2, 211–219.
• [14] V.A. Marchenko, Sturm–Liouville Operators and Applications, Oper. Theory: Adv. Appl., vol. 22, Birkhäuser Springer, 1986.
• [15] M. Möller, V. Pivovarchik, Direct and Inverse Finite-Dimensional Spectral Problems on Graphs, Oper. Theory: Adv. Appl., vol. 283, Birkhäuser Springer, 2020.
• [16] D. Mugnolo, V. Pivovarchik, Distinguishing co-spectral quantum graphs by scattering, J. Phys. A: Math. Theor. 56 (2023), no. 9, 095201.
• [17] M.-E. Pistol, Generating isospectral but not isomorphic quantum graphs, arXiv: 2104.12885.
• [18] V. Pivovarchik, On Ambarzumian type theorems for tree domains, Opuscula Math. 42 (2022), no. 3, 427–437.
• [19] V. Pivovarchik, Recovering the shape of an equilateral quantum tree by two spectra, Integral Equations Operator Theory 96 (2024), Article no. 11.
• [20] V. Pivovarchik, A. Chernyshenko, Cospectral quantum graphs with Dirichlet boundary conditions at pendant vertices, Ukrainian Math. J. 75 (2023), no. 3, 1–9.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023)