Abstract:
Free electron approximation for electron states in poly($p$-phenylene vinylene) and other conjugated systems is developed. It provides simple and clear analytical expressions for energies of electron states and for wave functions. The results are in quantitative agreement with experiments. Our model does not contain fitting parameters. We present two examples of the developed theory applications: exact calculation of electron energy in magnetic field, and scattering of electrons in copolymer.

Abstract:
We employ Wiegmann's solution of the Anderson impurity model in order to compute the compressibility of electron gas. We have found that there is a pair of neighbor levels separated by anomalously large energy $\propto L^{-1/3}$, where $L$ is the size of the system. The Fermi energy crosses this pseudogap and the compressibility decreases near the so-called ``symmetric'' limit.

Abstract:
All the principal results of this work concern the vertical transport in generic three dimensional superlattices. In the 1st chapter we make an historical introduction, then we discuss the geometry of the problem, and the physical parameters associated with structure of electron minibands and strength of external fields. We found the effect of collisionless transverse magnetoresistance, and we discuss it in the 2nd chapter. This effect is similar to collisionless Landau damping in a plasma and we utilize the same name. In the 3rd chapter we provide quantum mechanical reasons for the above effect; we show how a magnetic field bends narrow superlattice minibands, and we classify the states into Landau-type and Stark-type. In the 4th chapter we compute longitudinal magnetoresistance of superlattices due to the imperfections of the interfaces. Correlation length of the interface roughness can be measured independently by this method. In the 5th chapter we discuss the current-voltage characteristic of superlattice when an electric field destroys the one-miniband transport. We found that the structure of the high-field domains in the superlattices is complicated, but can be described analytically with great accuracy. This structure reveals itself in the details of the current-voltage characteristics. All our results are consistent with existing experiments, and we make careful comparison of theoretical predictions and experimental results in all chapters.

Abstract:
In this work we show that the Anderson impurity model applied to ``scar'' wave function may explain large fluctuations of ground state energy of electron gas in a quantum dot.

Abstract:
This is a three step work: i) we explain why quantum point contacts are similar to ballistic quantum dots; ii) we introduce the virtual Kondo state in both systems; iii-1st) this state explains 0.7 structure in point contacts; iii-2nd) formation of the local moment on this state is described by the nearly symmetric Anderson model, we solve it for finite size system having in mind quantum dots. We found one large level spacing $\Delta^\ast \propto (U\Gamma)^{1/2}\gg \Delta$, where $U$ is the charging energy of the virtual state, $\Gamma$ is the spectral width of this state and $\Delta$ is the mean level spacing of whole system. The theory explains periodicity of abnormal level spacing vs gate potential. The theory is in agreement with many experiments.

Abstract:
We suggest that the phonon dispersion in cuprates becomes strongly anisotropic due to interaction with spin waves; moreover the phonon dispersion becomes singular along $|k_x|=|k_y|$ directions. This would allow more electrons to form Cooper pairs and increase temperature of the superconducting transition. The interaction of phonons with spin waves is more important than the interaction of phonons with free electrons, because spin waves do not have the Fermi surface constrain.

Abstract:
The linear Zeeman effect in the molecular positronium Ps2 (dipositronium) is predicted for some of $S=1$, $M=\pm1$ states. This result is opposite to the case of the positronium atom; the latter has only quadratic Zeeman effect.

Abstract:
The decay rates of the density-density correlation function are computed for a chaotic billiard with some amount of disorder inside. In the case of the clean system the rates are zeros of Ruelle's Zeta function and in the limit of strong disorder they are roots of Selberg's Zeta function. We constructed the interpolation formula between two limiting Zeta functions by analogy with the case of the integrable billiards. The almost clean limit is discussed in some detail. PACS numbers: 05.20.Dd, 05.45.+b, 51.10.+y

Abstract:
Diagram, known in theory of the Anderson localization as the Hikami box, is computed for the Sinai billiard. This interference effect is mostly important for trajectories tangent to the opening of the billiard. This diagram is universal at low frequencies, because of the particle number conservation law. An independent parameter, which we call phase volume of diffraction, determines the corresponding frequency range. This result suggests that level statistics of a generic chaotic system is not universal.

Abstract:
In the present work we study the two-point correlation function $R(\epsilon)$ of the quantum mechanical spectrum of a classically chaotic system. Recently this quantity has been computed for chaotic and for disordered systems using periodic orbit theory and field theory. In this work we present an independent derivation, which is based on periodic orbit theory. The main ingredient in our approach is the use of the spectral zeta function and its autocorrelation function $C(\epsilon)$. The relation between $R(\epsilon)$ and $C(\epsilon)$ is constructed by making use of a probabilistic reasoning similar to that which has been used for the derivation of Hardy -- Littlewood conjecture. We then convert the symmetry properties of the function $C(\epsilon)$ into relations between the so-called diagonal and the off-diagonal parts of $R(\epsilon)$. Our results are valid for generic systems with broken time reversal symmetry, and with non-commensurable periods of the periodic orbits.