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.Comment: 22 pages (Latex file), 3 Postscript figures, to be published in Phys. Rev.B 55, N 20 (1997

We analyze the transport phenomena of two-dimensional quantum billiards with convex boundary of different shape. The quantum mechanical analysis is performed by means of the poles of the S matrix while the classical analysis is based on the motion of a free particle inside the cavity along trajectories with a different number of bounces at the boundary. The value of the conductance depends on the manner in which the leads are attached to the cavity. The Fourier transform of the transmission amplitudes is compared with the length of the classical paths. There is good agreement between classical and quantum mechanical results when the conductance is achieved mainly by special short-lived states such as whispering gallery modes and bouncing ball modes. In these cases, also the localization of the wave functions agrees with the picture of the classical paths. The S matrix is calculated classically and compared with the transmission coefficients of the quantum mechanical calculations for five modes in each lead. The number of modes coupled to the special states is effectively reduced.

It is shown that with the inclusion of the vertical extension of a quantum dot the experimental findings of Ashoori et al. [Phys. Rev. Lett. 71, 613 (1993)] can be modeled consistently with a parabolic confinement. Furthermore, the magnetic properties such as the magnetic moment and the susceptibility are sensitive to the presence and strength of a vertical confinement. Using a semiclassical approach the calculation of the eigenvalues reduces to simple quadratures providing a transparent and almost analytical quantization of the three-dimensional quantum dot energy levels that differ from the exact energies only by a few percent. While the dynamics for three-dimensional axially symmetric two-electron quantum dot with parabolic confinement potentials is in general nonseparable due to the Coulomb interaction we have found an exact separability for specific values of the magnetic field.