Abstract:
This short review presents a few case studies of finite electron systems for which strong correlations play a dominant role. In simple metal clusters, the valence electrons determine stability and shape of the clusters. The ionic skeleton of alkali metals is soft, and cluster geometries are often solely determined by electron correlations. In quantum dots and rings, the electrons may be confined by an external electrostatic potential, formed by a gated heterostructure. In the low density limit, the electrons may form so-called Wigner molecules, for which the many-body quantum spectra reveal the classical vibration modes. High rotational states increase the tendency for the electrons to localize. At low angular momenta, the electrons may form a quantum Hall liquid with vortices. In this case, the vortices act as quasi-particles with long-range effective interactions that localize in a vortex molecule, in much analogy to the electron localization at strong rotation.

Abstract:
In this review, we discuss the electronic structure of finite quantal systems on the nanoscale. After a few general remarks on the many-particle physics of the harmonic oscillator -- likely being the most studied example for the many-body systems of finite quantal systems, we discuss properties of metal clusters, quantum dots and cold atoms in traps. We address magic numbers, shape deformation, magnetism, particle localization, and vortex formation in rotating systems.

Abstract:
The configuration interaction (CI) method for calculating the exact eigenstates of a quantum-mechanical few-body system is problematic when applied to contact interactions between the particles. In two and three dimensions, the approach fails due to the pathology of the Dirac delta-potential, making it impossible to reach convergence by gradually increasing the size of the Hilbert space. However, for practical applications this problem may be cured in a rather simple manner, by renormalizing the STRENGTH of the contact potential, which must be diagonalized in a TRUNCATED Hilbert space. The procedure relies on the comparison of CI energies and wave functions with those obtained by the exact solution of the two-body Schrodinger equation for the regularized contact interaction. The rather simple scheme, while keeping the numerical procedures still elementary, nevertheless provides both cut-off-independent few-body physical observables and an estimate of the error of the CI calculation.

Abstract:
The purpose of this overview article, which can be viewed as a supplement to our previous review on quantum rings, [S. Viefers {\it et al}, Physica E {\bf 21} (2004), 1-35], is to highlight the differences of boson and fermion systems in one-dimensional (1D) and quasi-one-dimensional (Q1D) quantum rings. In particular this involves comparing their many-body spectra and other properties, in various regimes and models, including spinless and spinful particles, finite versus infinite interaction, and continuum versus lattice models. Our aim is to present the topic in a comprehensive way, focusing on small systems where the many-body problem can be solved exactly. Mapping out the similarities and differences between the bosonic and fermionic cases is of renewed interest due to the experimental developments in recent years, allowing for more controlled fabrication of both fermionic and bosonic quantum rings.

Abstract:
It was recently argued that in small quantum dots the electrons could crystallize at much higher densities than in the infinite two-dimensional electron gas. We compare predictions that the onset of spin polarization and the formation of Wigner molecules occurs at a density parameter $r_s\approx 4 a_B^*$ to the results of a straight-forward diagonalization of the Hamiltonian matrix.

Abstract:
We give a semiclassical interpretation of the mass asymmetry in the fission of heavy nuclei. Using only a few classical periodic orbits and a cavity model for the nuclear mean field, we reproduce the onset of left-right asymmetric shapes at the fission isomer minimum and the correct topology of the deformation energy surface in the region of the outer fission barrier. We point at the correspondence of the single-particle quantum states responsible for the asymmetry with the leading classical orbits, both lying in similar equatorial planes perpendicular to the symmetry axis of the system.

Abstract:
We discuss the localization of wavefunctions along planes containing the shortest periodic orbits in a three-dimensional billiard system with axial symmetry. This model mimicks the self-consistent mean field of a heavy nucleus at deformations that occur characteristically during the fission process [1,2]. Many actinide nuclei become unstable against left-right asymmetric deformations, which results in asymmetric fragment mass distributions. Recently we have shown [3,4] that the onset of this asymmetry can be explained in the semiclassical periodic orbit theory by a few short periodic orbits lying in planes perpendicular to the symmetry axis. Presently we show that these orbits are surrounded by small islands of stability in an otherwise chaotic phase space, and that the wavefunctions of the diabatic quantum states that are most sensitive to the left-right asymmetry have their extrema in the same planes. An EBK quantization of the classical motion near these planes reproduces the exact eigenenergies of the diabatic quantum states surprisingly well.

Abstract:
The yrast spectra (i.e. the lowest states for a given total angular momentum) of quantum dots in strong magnetic fields, are studied in terms of exact numerical diagonalization and analytic trial wave functions. We argue that certain features (cusps) in the many-body spectrum can be understood in terms of particle localization due to the strong field. A new class of trial wavefunctions supports the picture of the electrons being localized in Wigner molecule-like states consisting of consecutive rings of electrons, with low-lying excitations corresponding to rigid rotation of the outer ring of electrons. The geometry of the Wigner molecule is independent of interparticle interactions and the statistics of the particles.

Abstract:
We examine the rotational properties of a mixture of two Bose gases. Considering the limit of weak interactions between the atoms, we investigate the behavior of the system under a fixed angular momentum. We demonstrate a number of exact results in this many-body system.

Abstract:
We consider bosonic atoms that rotate in an anharmonic trapping potential. Using numerical diagonalization of the Hamiltonian, we identify the various phases of the gas as the rotational frequency of the trap and the coupling between the atoms are varied.