Mathematical Modelling and Geometry

8

Authors: A.N. Tsirulev and S.R. Usmanov

We consider complex scalar field minimally coupled to gravity and to electromagnetism. This note provides an examination of the spherically symmetric Einstein-Maxwell-Klein-Gordon system in a sufficiently general form supporting the maximal gauge freedom for metric functions and the electromagnetic potential. We study in detail gauge invariance and gauge transformations in the spherically symmetric case. Our method of reduction is based on the characteristic function, which is invariantly defined, up to sign, as the squared norm of the differential of the ’area’ metric function. For static configuration, one can extract a subsystem of three equations for the characteristic function, the electromagnetic potential, and the modulus of a scalar field.

18

Authors: Igor Filikhin, Sergei Matinyan and Branislav Vlahovic

We study electron localization and tunneling in double quantum dots (DQD) and rings (DQR). The wave function of a single electron may be localized in one of the QDs or be delocalized when is spread over the whole system. Spectral distributions of such states are considered. Electron transition between both states is possible if inter-dot distance or QD (QR) geometry is varied. The eect of violation of symmetry of DQDs geometry on the tunneling is studied in details. We show that a small violation of symmetry drastically aects localization of electron. The same symmetry violation eect happens if an electrical or magnetic eld are applied. We consider anti-crossing of the levels as the mechanism of the tunneling between the localized and delocalized states.