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Information × Registration Number 0225U005277, (0124U000818) , R & D reports Title Inverse problems of finding the shape of a graph by spectral data popup.stage_title Розробка методів знаходження форми графа, виходячи з відомої S-функції розсіяння. Head Pyvovarchyk Viacheslav M., д.ф.-м.н. Registration Date 29-12-2025 Organization The State Institution “South Ukrainian National Pedagogical University named after K. D. Ushynsky” popup.description1 The goal of the project is to use estimates from above for possible multiplicities of eigenvalues of finite-dimensional spectral problems to describe normal eigenvalues of problems from quantum graph theory, in particular, obtaining analogs of Bargman's inequality for spectral problems on graphs. This project will also consider cospectral quantum graphs, which should exist in the case of a sufficiently large number of vertices. It will also be determined whether the shape of the graph can be uniquely determined if the potentials on the edges are not zero. popup.description2 The object of the study is the so-called "quantum graphs", that is, problems generated by the differential equations of quantum mechanics, in particular Schrödinger, Sturm-Liouville and Dirac on domains that are graphs. When considering the scattering problem on a metric tree formed by the connection of a semi-infinite edge to an equilateral graph, the fundamental theory of V.A. Marchenko was used to describe the analytic properties of the S-scattering function. Provided that the equation on the semi-infinite edge is not perturbed, it is proved that the Jost functions are entire functions of exponential type in the variable √λ where λ is the spectral parameter, and the S-function is meromorphic. It is found that such a problem corresponds to a self-adjoint operator bounded from below and has an essential continuous spectrum that lies on the non-negative semi-axis of the λ-plane and can have eigenvalues imbedded in the continuous spectrum, and isolated eigenvalues of finite multiplicity. It is proved that the S-function of such a problem can be expressed in terms of the characteristic functions of the spectral Dirichlet and Neumann problems given on a compact subgraph, described in the results of the previous stage. All simple connected equilateral graphs with the number of edges not exceeding 7 are considered. It is proved that the problem for a compact subgraph with standard conditions at internal vertices and Dirichlet conditions at pendant vertices is uniquely determined by its spectrum. All obtained results are new. Most of the previous results on the search for co-spectral graphs were related to classical graph theory, while the research of this stage of the project belongs to quantum graph theory. The research in this project is theoretical. All the results obtained are scientifically substantiated and proven. They are based on the axioms of mathematics. Product Description popup.authors Boyko Olga Pavlovna Kaliuzhnyi-Verbovetskyi Dmytro S. Dudko Anastasiia Ihorivna Olha M. Boldarieva popup.nrat_date 2025-12-29 Close