On Ambarzumian type theorems for tree domains

• Autorzy: Pivovarchik Vyacheslav

Identyfikatory

DOI

10.7494/OpMath.2022.42.3.427

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Sturm-Liouville equation; eigenvalue; equilateral tree; star graph; Dirichlet boundary condition; Neumann boundary condition

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2022
• Tom: Vol. 42, no. 3
• Strony: 427--437
• Opis fizyczny: Bibliogr. 20 poz.

Twórcy

autor

Pivovarchik Vyacheslav

• South-Ukrainian National Pedagogical University, Staroportofrankovskaya Str. 26, Odessa, 65020, Ukraine

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

Bibliografia

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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).