The review considers the peculiarities of symmetry breaking and symmetry transformations and the related physical effects in finite quantum systems. Some types of symmetry in finite systems can be broken only asymptotically. However, with a sufficiently large number of particles, crossover transitions become sharp, so that symmetry breaking happens similarly to that in macroscopic systems. This concerns, in particular, global gauge symmetry breaking, related to Bose-Einstein condensation and superconductivity, or isotropy breaking, related to the generation of quantum vortices, and the stratification in multicomponent mixtures. A special type of symmetry transformation, characteristic only for finite systems, is the change of shape symmetry. These phenomena are illustrated by the examples of several typical mesoscopic systems, such as trapped atoms, quantum dots, atomic nuclei, and metallic grains. The specific features of the review are: (i) the emphasis on the peculiarities of the symmetry breaking in finite mesoscopic systems; (ii) the analysis of common properties of physically different finite quantum systems; (iii) the manifestations of symmetry breaking in the spectra of collective excitations in finite quantum systems. The analysis of these features allows for the better understanding of the intimate relation between the type of symmetry and other physical properties of quantum systems. This also makes it possible to predict new effects by employing the analogies between finite quantum systems of different physical nature.

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.

Classical and quantum mechanical results are reported for the single particle motion in a harmonic oscillator potential which is characterized by a quadrupole deformation and an additional octupole deformation. The chaotic character of the motion is strongly dependent on the quadrupole deformation in that for a prolate deformation virtually no chaos is discernible while for the oblate case the motion shows strong chaos when the octupole term is turned on.

The physics of nanoscale systems has advanced rapidly over the last few years. A consistent description of these small systems is a challenging task for quantum theory since their properties may be influenced strongly by attaching leads to them [1][2][3][4][5][6][7][8][9]. They are simulated often by means of quantum billiards. When the cavity is not fully opened, the propagation of the mode is restricted to energies at which the overlap integral between the wave functions of the resonance states and the channel modes has a non-vanishing value. In the case of well isolated resonances, the electron can propagate therefore only at the energies of the resonance states ("resonance tunneling"). Due to the coupling between the internal states of the cavity and the channel mode, the states get widths. When the coupling is sufficiently strong, the resonances start to overlap and to interact via the channel mode.As a consequence, some redistribution in the resonance states of the cavity takes place. It 1

Recent studies of transport phenomena with complex potentials are explained by generic square root singularities of spectrum and eigenfunctions of non-Hermitian Hamiltonians. Using a two channel problem we demonstrate that such singularities produce a significant effect upon the pole behaviour of the scattering matrix, and more significantly upon the associated residues. This mechanism explains why by proper choice of the system parameters the resonance cross section is increased drastically in one channel and suppressed in the other channel.

We present a dynamical analysis of the transport through small quantum cavities with large openings. The systematic suppression of shot noise is used to distinguish direct, deterministic from indirect, indeterministic transport processes. The analysis is based on quantum mechanical calculations of $S$ matrices and their poles for quantum billiards with convex boundaries of different shape and two open channels in each of the two attached leads. Direct processes are supported when special states couple strongly to the leads, and can result in deterministic transport as signified by a striking system-specific suppression of shot noise.Comment: 4 pages, 3 figure

While the dynamics for three-dimensional axially symmetric two-electron quantum dots with parabolic confinement potentials is in general nonseparable we have found an exact separability with three quantum numbers for specific values of the magnetic field. Furthermore, it is shown that 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 quantum dot energy levels which differ from the exact energies only by a few percent.PACS numbers: 73.21. La, 03.65.Sq, 75.75.+a, 05.45.Mt Current nanofabrication technology allows one to control the size and shape of quantum dots [1][2][3]. Due to the confinement of the electrons in all three spatial directions the energy spectrum is quantized creating excellent experimental and theoretical opportunities to study controlled single-particle and collective dynamics at the atomic scale. For example, depending on the experimental setup, the spectrum of a quantum dot displays shell structure

Using a classical and quantum mechanical analysis, we show that the magnetic field gives rise to dynamical symmetries of a three-dimensional axially symmetric two-electron quantum dot with a parabolic confinement. These symmetries manifest themselves as near-degeneracies in the quantum spectrum at specific values of the magnetic field and are robust at any strength of the electron-electron interaction. The Hamiltonian of a two-electron QD in a magnetic field readswhere α = e 2 /4πε 0 ε r . Here, e, m * , ε 0 and ε r are the unit charge, effective electron mass, vacuum and relative dielectric constants of a semiconductor, respectively.For the perpendicular magnetic field we choose the vector potential with a gauge A =