Registration Number
0216U005414,
0115U002671
, R & D reports
Title
Spectral properties of non-classical models of quantum mechanics
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Head
Man'ko S.S.,
Registration Date
26-01-2016
Organization
Pidstryhach Institute of Applied Problems of Mechanics and Mathematics NAS of Ukraine
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Object of the research - ordinary differential operators of mathematical physics with singular coefficients and/or domains. Purpose of the research - to study the spectral properties of the Dirac and (magnetic) Schr?dinger operators which have periodic and/or singular coefficients. The research methods - methods of the theory of differential equations and spectral theory of self-adjoint operators as well as asymptotic methods. Main research results: - We have studied the kvantum mechanical particles in periodic structures under the influence of periodic magnetic fluxes. Periodic problems with complex geometry arises in the study of advanced materials (such as optical crystals, graphene and other carbon nanostructures). Mathematical fundament used in the study of these models is based on the general Floquet-Bloch theory as well as on the perturbation theory of self-adjoint operators. - Another important result of research concerns the inverse scattering problem for differential operators of Schr?dinger and Dirac types. We extended the classical scattering theory (due to Marchenko) to classes of singular potentials in the case of the Schr?dinger operators and for classes of integrable potentials in the case of the Dirac operators. This research provides possibilities for solving the Cauchy problem with singular initial data for many nonlinear equations (Korteweg-de Vries equation, the nonlinear Schrodinger equation, etc.).
Product Description
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Манько С.С.
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2020-04-02
