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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.

Higher order Nevanlinna functions and the inverse three spectra problem

Boyko O., Martinyuk O., Pivovarchik V.

Opuscula Mathematica

2016

Vol. 36, no. 3

301--314

he three spectra problem of recovering the Sturm-Liouville equation by the spectrum of the Dirichlet-Dirichlet boundary value problem on [0, α], the Dirichlet-Dirichlet problem on [0, α/2] and the Neumann-Dirichlet problem on [α/2, α] is considered. Sufficient conditions of solvability and of uniqueness of the solution to such a problem are found.