We calculate the energy level statistics in a two-dimensional disc with diffusive boundary scattering by the means of the recently proposed ballistic nonlinear sigma-model.

A new configuration of the random potential and the optimal fluctuation method are used for the evaluation of the long-time delay in Ohmic response of a finite piece of a disordered one-dimensional chain. The result agrees, to exponential accuracy, with that obtained earlier by Altshuler and Prigodin.

We suggest an effective field theory for disorderd conductors, which describes quantum kinetics of ballistically propagating electrons. This theory contains non-linear $\sigma$-model \cite{Efetov} as its long wave limit.

The spectrum of a two-dimensional electron system in a square lattice potential subject to a perpendicular magnetic field is considered. Well known symmetry considerations show that if the magnetic field flux per unit cell (measured in units of the flux quantum hc/ mod e mod ) is rational and non-integer, the electron energy can be expressed as a function of a generalized momentum k and the spectrum consists of translationally degenerate bands in k-space. In this work it is shown that if the chemical potential is located at a logarithmic Van Hove singularity of the density of states then it is possible that a periodic lattice distortion is energetically favourable. This instability persists when the flux is changed within a small interval. The width of this interval as well as the critical temperature marking the onset of the instability can be estimated with logarithmic accuracy. The effects of electron-electron interactions are considered in the Hartree approximation.

The distribution function for the current noise in a normal-metal-superconductor quantum point contact is calculated. It is shown that this distribution describes independent processes when a charge +/-e0 or +/-2e0 passes through the contact. At low temperature and voltage only processes with double-charge transfer are relevant. At zero temperature and low voltage the distribution has a binomial form.

We study the universal conductance fluctuations (UCF) in two-dimensional mesoscopic systems in the presence of a strong magnetic field when the Hall conductance σxy is not negligible compared to σxx. We find that the Hall component of the diffusion current modifies the boundary conditions for the diffuson and gives rise to a qualitatively new behavior of the UCF. In particular, when σxy is larger than σxx, the magnetic field correlation length Bc is proportional to σxx instead of the usual inverse proportionality leading to oscillations in Bc in phase with the Shubnikov-de Haas oscillations of σxx.

We discuss the distribution function for the current noise in quantum point contacts. Special interest is paid to a contact of a superconductor with a normal metal. A new derivation of the Lesovik-Levitov formulae is suggested. It is shown, for the SN contacts, that the distribution of the noise describes independent processes when charge ±e0or ±2e0 passes through the contact. At low temperature and voltage only processes with double charge transfer are relevant. At zero temperature and low voltage the distribution has a binomial form.

In the framework of the scaling hypothesis of localisation of 2D electrons in a magnetic field B and random potential it is shown that a discrete set of extended states, characteristic for Quantum Hall Effect, does not disappear at vanishing B, when Landau levels overlap. Energies of these states E(n) in weak fields did not follow Landau levels going to zero, but, contrary, are increasing for decreasing field, floating up like bubbles and leaving Fermi sea. As a result, at weak enough magnetic fields all states below chemical potential are localised. For case of two overlapped Landau levels (two layers system or two overlapped spin subbands) a new phenomenon of spontaneous splitting of extended states is predicted.

Fluctuations of the current-voltage (IV) characteristic of a mesoscopic sample are considered. Such a sample can be realized as a junction of two normal metals. The voltage dependence of current is a random function, and the IV-characteristic has a "blades of grass" form on a usual ohmic background. The scale of each "blade" in voltage is of order Vc ~ ℏ/τfe, where τf is the time of flight through the junction and e is the electron charge. The scale in current δI depends on the voltage V and temperature T, and of order (e2/ℏ)(VVc)1/2, if V ≫ Vc, T/e. As a result, provided the voltage is sufficiently large, there appear regions of negative differential resistance.

The stationary Josephson effect in S-N-S junction is considered. It is found that except the usual term there is another contribution to the supercurrent which is due to the interaction between the electrons in the normal metal. The features of this extra supercurrent are: 1) it is proportional to the electron-electron coupling constant so its sign is arbitrary, 2) the period of its dependence on phase discontinuity is π (not 2π as usual). This additional supercurrent will dominate the usual one if normal metal is in the ferromagnetic state. Therefore, if the electrons in ferromagnet repulse each other the phase discontinuity is equal to π /2 in the ground state. A system with finite current in the ground state can be constructed.