Table of contents

A nonlinear change of basis allows us to show that the non-standard quantum deformation of the (3 + 1) Poincaré algebra has a bicrossproduct structure. Quantum universal R-matrix, Pauli - Lubanski and mass operators are presented in the new basis.

We present a general theory of quasiprobability distributions on phase spaces of quantum systems whose dynamical symmetry groups are (finite-dimensional) Lie groups. The family of distributions on a phase space is postulated to satisfy the Stratonovich - Weyl correspondence with a generalized traciality condition. The corresponding family of the Stratonovich - Weyl kernels is constructed explicitly. In the presented theory we use the concept of generalized coherent states, that brings physical insight into the mathematical formalism.

We consider the dynamics of a model introduced recently by Bialas, Burda and Johnston. At equilibrium the model exhibits a transition between a fluid and a condensed phase. For long evolution times the dynamics of condensation possesses a scaling regime that we study by analytical and numerical means. We determine the scaling form of the occupation number probabilities. The behaviour of the two-time correlations of the energy demonstrates that aging takes place in the condensed phase, while it does not in the fluid phase.

On-line learning of a rule given by an N-dimensional Ising perceptron is considered for the case when the student is constrained to take values in a discrete state space of size . For L = 2 no on-line algorithm can achieve a finite overlap with the teacher in the thermodynamic limit. However, if L is on the order of , Hebbian learning does achieve a finite overlap.

The class of quantum integrable systems associated with root systems was introduced as a generalization of the Calogero - Sutherland systems. In this letter, a new property of such systems is proved to be valid. Namely, in the case of the potential , the series for the product of two wavefunctions coincides with the Clebsch - Gordan series. This gives the recursive relations for the wavefunctions of such systems and for generalized spherical functions related to them on symmetric spaces.

One conjectures that the Clebsch - Gordan series is also unchanged under more general two-parametric deformation (-deformation).

Let q(r), r=|x|, , be a real-valued square-integrable compactly supported function, and be the smallest interval containing the support of q(r). Let be the corresponding scattering amplitude at a fixed positive energy, . Let be the phase shifts at k = 1. It is proved that , provided that q(r) does not change sign in some, arbitrary small, neighbourhood of a.

Elementary excitations in the one-dimensional Hubbard model with boundary fields are discussed. Boundary scattering matrices for the excitations of the charge and spin sectors are evaluated. Both the repulsive and the attractive Hubbard models at half-filling and without a magnetic field are studied.

Dynamical ensemble equivalence between hydrodynamic dissipative equations and suitable time-reversible dynamical systems has been investigated in a class of dynamical systems for turbulence. The reversible dynamics is obtained from the original dissipative equations by imposing a global constraint. We find that, by increasing the input energy, the system changes from an equilibrium state to a non-equilibrium stationary state in which an energy cascade, with the same statistical properties of the original system, is clearly detected.

A relativistic generalization of the quantum-state diffusion model is developed. The model describes a Dirac electron which is coupled to an external electromagnetic field and a dissipative environment. A relativistically covariant stochastic Dirac equation is obtained by regarding the state vector as a functional on a certain set of spacelike hypersurfaces in Minkowski space and by the definition of an appropriate Hilbert bundle on this set. The integrability condition of the stochastic process and the corresponding covariant density matrix equation are derived. Further, the relativistic equations governing the dynamical state-vector localization are deduced.

We compute by means of the Bethe ansatz the boundary S-matrix for the open anisotropic spin- chain with diagonal boundary magnetic fields in the noncritical regime . Our result, which is formulated in terms of q-gamma functions, agrees with the vertex-operator result of Jimbo et al.

We investigate the anisotropic integrable spin chain consisting of spins and s = 1 by means of thermodynamic Bethe ansatz for the anisotropy , where the analysis of the Takahashi conditions leads to a more complicated string picture. We give the phase diagram with respect to the two real coupling constants and , which contain a new region where the ground state is formed by strings with infinite Fermi zones. In this region the velocities of sound for the two physical excitations have been calculated from the dressed energies. This leads to an additional line of conformal invariance not known before.

In this paper we propose and solve analytically a class of models generalizing the ballistic deposition (BD) model for irreversible adsorption of hard particles onto a line. In these new models, when an incoming particle interacts with an adsorbed one, it adsorbs next to it leaving between them a small gap of random size with characteristic length . The first moments of the gap distribution suffice to describe the lowest-order corrections to the BD model. In particular, for small values of we obtain an asymptotic approach to the BD jamming coverage in agreement with previous Brownian simulations which take into account diffusion and gravity.

We present an analytic real-space renormalization group calculation for the random-field Ising model. We apply the Migdal-Kadanoff approximation for the renormalization of a cubic cell in dimensions d, introducing a new field partitioning scheme which allows us to treat the random-field fluctuations in a coherent manner. Our scheme leads naturally to a lower critical dimensionality and allows us to calculate a complete set of three independent exponents in arbitrary dimension. In three dimensions the magnetization exponent and the Schwartz-Soffer inequality is almost satisfied as an equality. We expand analytically in . Further, we show that and the magnitude of the inequality go to zero exponentially with . We calculate the crossover exponent, from pure to the random-field system and find surprisingly good agreement with experimental values. We find that satisfies the Schwartz-Soffer inequality: , the susceptibility exponent of the pure system. We expand in and find that the magnitude of the inequality varies exponentially in . Finally we find that dimensional reduction is satisfied to first order in , with the reduced dimension .

We introduce a theoretical model for the compaction of granular materials by discrete vibrations which is expected to hold when the intensity of vibration is low. The dynamical unit is taken to be clusters of granules that belong to the same collective structure. We rigourously construct the model from first principles and show that numerical solutions compare favourably with a range of experimental results. This includes the logarithmic relaxation towards a statistical steady state, the effect of varying the intensity of vibration resulting in a so-called `annealing' curve, and the power spectrum of density fluctuations in the steady state itself. A mean-field version of the model is introduced which shares many features with the exact model and is open to quantitative analysis.

In this paper, we study the generalization ability of a simple perceptron which learns an unrealizable Boolean function represented by a perceptron with a non-monotonic transfer function of reversed-wedge type. This type of non-monotonic perceptron is considered as a variant of multilayer perceptron and is parametrized by a single `wedge' parameter a. Reflecting the non-monotonic nature of the target function, a discontinuous transition from the poor generalization phase to the good generalization phase is observed in the learning curve for intermediate values of a. We also find that asymptotic learning curves are classified into the following two categories depending on a. For large a, the learning curve obeys a power law with exponent 1. On the other hand, a power law with exponent is obtained for small a. Although these two exponents are obtained from unstable replica symmetric solutions by using the replica method, they are consistent with the results obtainable without using the replica method in a low-dimensional version of this learning problem. This suggests that our results are good approximations even if they are not exact.

We numerically study the statistics of the transport parameters of the system of disordered chains, mutually coupled by hopping term . We find that the system can be described in terms of the random matrix theory with t-dependent `symmetry parameter' . In the limit behaves as with .

We consider the multifractal properties of a quasiperiodic tight binding Hamiltonian where the hopping elements are arranged according to the Fibonacci chain. By using the trace map approach and an assumption relating the cycles of the trace map to the integrated density of states, it is shown that the maximal scaling of the spectrum does not always occur at the edges or the centre of the spectrum, which is confirmed by numerical simulations. A good description of the multifractal spectrum is obtained by applying the 3z-model which was recently developed for the Harper model.

We derive a uniform approximation for semiclassical contributions of periodic orbits to the spectral density which is valid for generic period-quadrupling bifurcations in systems with a mixed phase space. These bifurcations involve three periodic orbits which coalesce at the bifurcation. In the vicinity of the bifurcation the three orbits give a collective contribution to the spectral density while the individual contributions of Gutzwiller's type would diverge at the bifurcation. The uniform approximation is obtained by mapping the action function onto the normal form corresponding to the bifurcation. This article is a continuation of previous work in which uniform approximations for generic period-m-tupling bifurcations with were derived.

We solve the cooperative sequential adsorption problem on a linear chain consisting of periodically repeating sequences of sites with different adsorption rates, patches, for example The problem is reduced to a system of s first-order differential equations, where s is the patch size. A simple iterative method to solve the equations numerically is presented. A few examples are calculated in detail.

We show that new universality of Lyapunov spectra exists in Hamiltonian systems with many degrees of freedom. The universality appears in systems which are neither nearly integrable nor fully chaotic, and it is different from the one which is obtained in fully chaotic systems on one-dimensional chains as follows. One is that the universality is found in a finite range of large rather than the whole range, where N is the number of degrees of freedom. Another is that Lyapunov spectra are not straight, while fully chaotic systems give straight Lyapunov spectra even on the three-dimensional simple cubic lattice. The universality appears when quadratic terms of a potential function dominate higher terms, harmonic motions are hence regarded as the base of global motions.

The general structure of kink manifolds in (1 + 1)-dimensional complex scalar field theory is described by analysing three special models. New solitary waves are reported. Kink energy sum rules arise between different types of solitary waves.

Covariant stochastic partial differential equations (SPDEs) are studied in any dimension. A special class of such equations is selected and it is proved that the solutions can be analytically continued to Minkowski spacetime yielding tempered Wightman distributions which are covariant, obey the locality axiom and a weak form of the spectral axiom.

Approximate solutions of matrix linear differential equations by matrix exponentials are considered. In particular, the convergence issue of Magnus and Fer expansions is treated. Upper bounds for the convergence radius in terms of the norm of the defining matrix of the system are obtained. The very few previously published bounds are improved. Bounds to the error of approximate solutions are also reported. All results are based just on algebraic manipulations of the recursive relation of the expansion generators.

The mechanism for the generation of multipole moments due to an external field is presented for the Born-Infeld charged particle. The `polarizability coefficient' for arbitrary l-pole moment is calculated. It turns out that , where and b is the Born-Infeld nonlinearity constant. Some physical implications are considered.

The main aim of this paper is to realize that it is feasible to construct a `periodic orbit theory' of localization by extending the idea of classical action correlations. This possibility had been questioned by many researchers in the field of `quantum chaos'. Starting from the semiclassical trace formula, we formulate a quantal-classical duality relation that connects the spectral properties of the quantal spectrum to the statistical properties of lengths of periodic orbits. By identifying the classical correlation scale it is possible to extend the semiclassical theory of spectral statistics, in case of a complex systems, beyond the limitations that are implied by the diagonal approximation. We discuss the quantal dynamics of a particle in a disordered system. The various regimes are defined in terms of time-disorder `phase diagram'. As expected, the breaktime may be `disorder limited' rather than `volume limited', leading to localization if it is shorter than the ergodic time. Qualitative agreement with scaling theory of localization in one to three dimensions is demonstrated.

We show that the calculation of Berezin integrals over anticommuting variables can be reduced to the evaluation of expectations of functionals of Poisson processes via an appropriate Feynman-Kac formula. In this way the tools of ordinary analysis can be applied to Berezin integrals and, as an example, we prove a simple upper bound. Possible applications of our results are briefly mentioned.

A class of magnetic control operations permitting one to achieve the Brown and Carson effect (shrinking or expansion of the wavepacket in coordinate space) by applying sinusoidal magnetic pulses is presented, with numerical data. An analogue of the Strutt diagramm for the phenomenon is obtained. We show that the scale operation is achieved as a `boomerang effect'; its repetitions (when applying periodic magnetic pulses), cannot assure the `monotonic tracking' of the packet size to zero. This is due to a new `no go argument' which forbids the generation of the multiple squeezing effects without paying a price in the form of intermediate expansions.

We study in detail the bound-state spectrum of the generalized Morse potential (GMP), which was proposed by Deng and Fan as a potential function for diatomic molecules. By connecting the corresponding Schrödinger equation with the Laplace equation on the hyperboloid and the Schrödinger equation for the Pöschl-Teller potential, we explain the exact solvability of the problem by an symmetry algebra, and obtain an explicit realization of the latter as . We prove that some of the generators connect among themselves wavefunctions belonging to different GMPs (called satellite potentials). The conserved quantity is some combination of the potential parameters instead of the level energy, as for potential algebras. Hence, belongs to a new class of symmetry algebras. We also stress the usefulness of our algebraic results for simplifying the calculation of Frank-Condon factors for electromagnetic transitions between rovibrational levels based on different electronic states.

We point out that in certain cases, all the differential equations (for given indipendent and dependent variables) possessing a given symmetry necessarily share a common solution. Under weaker conditions, all such differential equations have a solution - in general, different for different equations - characterized by a common symmetry. We characterize this situation and the common solution, or the common symmetry of solutions, and give concrete examples.

We consider the three-dimensional three-wave resonance equation using a bilinear approach to investigate a broad class of solutions. Solutions are obtained in a Grammian form, and their relationship to Kaup's solutions examined.

When using the Born-Oppenheimer approximation for molecular systems, one encounters a quantum mechanical Hamiltonian for the electrons that depends on several parameters that describe the positions of the nuclei. As these parameters are varied, the spectrum of the electron Hamiltonian may vary. In particular, discrete eigenvalues may approach very close to one another at `avoided crossings' of the electronic energy levels. We give a definition of an avoided crossing and classify generic avoided crossings of minimal multiplicity eigenvalues. There are six distinct types that depend on the dimension of the nuclear-configuration space and on the symmetries of the electron Hamiltonian function.

A quantum-mechanical particle kicked by a rank 1 perturbation is a solvable model of a scattering system with a time periodic Hamiltonian. We study its spectral properties and compute explicitly the scattering quantities. Below a critical value, there are certain ranges of periods for which the system has stable states under the time evolution (cyclic states). These cyclic states lose their stability as the period increases and may be transformed into resonances. This is an example of the general phenomenon of stabilization of quantum states under a high-frequency perturbation.

In ordinary and relativistic quantum mechanics the energy spectra of most of the Hamiltonians cannot be obtained exactly. Approximate methods have to be used among which the variational one is particuarly popular. The purpose of this paper is to show that when the matrices appearing in a Dirac equation with interaction, are replaced by a direct product of matrices associated with ordinary and what we call sign spins, then a standard complete set of non-relativistic wavefunctions can be used to carry out the variational calculations. To illustrate the power of our method we analyse first the variational energies of ordinary and Dirac relativistic oscillator Hamiltonians, and then indicate the procedure for the general one-particle case. The extensions to higher spins, or to a larger number of particles, are briefly mentioned in the conclusion.

A method for solving the Schwinger-Dyson equations for the Green function generating functional of non-Abelian gauge theory is proposed. The method is based on an approximation of the Schwinger-Dyson equations by exactly soluble equations. For the SU(2) model the first-step equations of the iteration scheme are solved which define a gauge field propagator. Apart from the usual perturbative solution, a non-perturbative solution is found, which corresponds to the spontaneous symmetry breaking and obeys the infrared finite behaviour of the propagator.