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Information × Registration Number 0224U033597, (0124U000818) , R & D reports Title Inverse problems of finding the shape of a graph by spectral data popup.stage_title Розробка методу знаходження форми графу, виходячи з асимптотики власних значень для зв’яних графів. Head Pyvovarchyk Viacheslav M., д.ф.-м.н. Registration Date 31-12-2024 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 investigation is the so called ‘quantum graphs’, i.e. problems generated by differential equations of quantum mechanics in particular Schroedinger, Sturm-Liouville and Dirac equations on quasi-one-dimensional graph domains. Another object is investigation of the connection between finite-dimensional and infinite-dimensional problems on graphs, in particular on trees. On this stage we used algebraic approach to problems generated by differential equations. The asymptotics of the characteristic functions and of the eigenvalues were obtained. It was shown how to find the shape of a graph using the knowledge of the leading terms of the eigenvalue asymptotics (at least for small number of vertices). It was shown also that even if the leading terms of the asymptotics correspond to two non-isomorphic graphs the next terms help to determine uniquely the shape of the graph. All the obtained results are new. The majority of the prior results corresponding to co-spectral graphs were obtained in approach of classical graph theory while the results of this stage describe this question in quantum graph theory. The investigations of this project are theoretical. The using of the two spectra of boundary value problems to find the shape of a metric tree open new horizon in this field. All the obtained results are scientifically justified and proved. They are based on axioms of mathematics. Since the project is theoretical, the obtained results are valuable for the needs of development of the country and for universal community in future. The results of this project can be used in synthesis of electrical circuits and in design of micro-schemes. They represent essential contribution into development of domestic and world science. Product Description popup.authors Boiko Olha P. Boldarieva Olha M. Voronenko Oleksandr V. Dudko Anastasiia I. Kaliuzhnyi-Verbovetskyi Dmytro S. popup.nrat_date 2024-12-31 Close