TY  - JOUR
ID  - Pivovarchik2022
LB  - Pivovarchik2022
AU  - Pivovarchik, Vyacheslav
TI  - On Ambarzumian type theorems for tree domains
ST  - On Ambarzumian type theorems for tree domains
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2022
VL  - 42
IS  - 3
SP  - 427
EP  - 437
DO  - https://doi.org/10.7494/OpMath.2022.42.3.427
UR  - https://doi.org/10.7494/OpMath.2022.42.3.427
AB  - 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.
KW  - Sturm-Liouville equation
KW  - eigenvalue
KW  - equilateral tree
KW  - star graph
KW  - Dirichlet boundary condition
KW  - Neumann boundary condition
SN  - 1232-9274
ER  -