Two interacting electrons in a harmonic oscillator potential under the influence of a perpendicular homogeneous magnetic field are considered. Analytic expressions are obtained for the energy spectrum of the two-and three-dimensional cases. Exact conditions for phase transitions due to the electron-electron interaction in a quantum dot as a function of the dot size and magnetic field are calculated.
Crossings between spin-singlet and spin-triplet lowest states are analyzed within the model of a two-electron quantum dot in a perpendicular magnetic field. The explicit expressions in terms of the magnetic field, the magnetic quantum number m of the state and the dimensionless dot size for these crossings are found.
Based on the investigation of the asymptotic behaviour of the polarization loop function for charged n scalar particles in an external gauge field, we determine the interaction Hamiltonian including relativistic corrections. The mass spectrum of the bound state is analytically derived. The mechanism for arising of the constituent mass of the relativistic bound-state forming particles is explained. The mass and the constituent mass of the two-, three- and n-body relativistic bound states are calculated taking into account relativistic corrections. The corrections arising due to the one- and two-loop electron polarization to the energy spectrum of muonic hydrogen with orbital and radial excitations are calculated.
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