Table of contents

We consider a model in which positive and negative particles diffuse in an asymmetric, CP-invariant way on a ring. The positive particles hop clockwise, the negative counter-clockwise and oppositely-charged adjacent particles may swap positions. Monte Carlo simulations and analytic calculations suggest that the model has three phases; a `pure' phase in which one has three pinned blocks of only positive, negative particles and vacancies, and in which translational invariance is spontaneously broken, a `mixed' phase with a non-vanishing current in which the three blocks are positive, negative and neutral, and a disordered phase without blocks.

A continuous cyclic sequence of quantum states has an associated geometric, or Berry, phase . For spin J, such a sequence is described by a cyclic change in the 2J+1 coefficients of the basis states . The Berry phase is analysed here for the general case - that is, the coefficients are allowed to vary in an arbitrary cyclic manner. The result is expressed in geometric terms, specifically in the democratic representation due to Majorana. This uniquely characterizes the spin state , up to overall phase, by the positions of 2J dots on the unit sphere of directions in real space. If the positions are denoted by unit vectors , where , each traces out a parametrized loop on the sphere, and the Berry phase is given by an integral of combinations of these vectors.

The general form is given for a force acting on a vortex in the ferromagnet model without any assumption for the profile of the vortex. The strength of the force is shown to be written in terms of the Mermin - Ho topological invariant for the vortex in superfluid He3-A. The discussions are given on the point of how the resulting formula can be interpreted as the mapping degree for the vortex configuration. These results are extended to the system described by the generalized spin.

A general formalism is described which allows the study of a wide range of stochastic problems related to evolution. We apply it to the treatment of the evolution of a finite population in the presence of mutations and selective advantage. The results are close to those of Eigen and Schuster's error threshold model, in the presence of randomly drawn sterile sequences. The theoretical results are in excellent agreement with numerical simulations.

An alternative to the Lüders generalization of the von Neumann projection is formulated allowing a generalization to measurements represented by positive operator-valued measures. This generalization is based on the notions of non-orthogonal projections and bi-orthonormal systems in Hilbert-Schmidt space, and is closely related to phase-space representations of quantum mechanics. A number of applications will be discussed. It is demonstrated that the alternative projection has a natural interpretation as a representation of the information on the initial state, to be obtained by the measurement.

Kolmogorov-Arnold-Moser boundaries appear in sufficiently smooth two-dimensional area-preserving maps. When such boundaries are destroyed, they become pseudo-boundaries. We show that pseudo-boundaries can also be found in discontinuous maps. These pseudo-boundaries originate in groups of chains of islands which separate parts of the phase space and need to be crossed in order to move between the different subspaces. Trajectories, however, do not easily cross these chains, but tend to propagate along them. This type of behaviour is demonstrated by using a `generalized' Fermi map.

A perturbative description of photodetection without the rotating wave approximation is presented. With the help of Green function techniques on an extended time contour renormalized expressions for the single- and multitime counting probabilities are derived, which explicitly reflect the causal propagation of signals. In the case of classical light fields the moments of the counting statistics can be expressed in terms of field correlation functions in agreement with the result of Bykov and Tatarskii. For non-classical radiation fields such as squeezed states the Bykov-Tatarskii formula is shown to be incorrect.

The Drude weight for the one-dimensional Hubbard model is investigated at finite temperatures by using the Bethe ansatz solution. Evaluating finite-size corrections to the thermodynamic Bethe ansatz equations, we obtain the formula for the Drude weight as the response of the system to an external gauge potential. We perform low-temperature expansions of the Drude weight in the case of half-filling as well as away from half-filling, which clearly distinguish the Mott-insulating state from the metallic state.

Recently we have found identical behaviour for various spin models on `thin' random graphs - Feynman diagrams - and the corresponding Bethe lattices. In this paper we observe that the ratios of the saddle-point equations in the random graph approach are identical to the fixed point(s) of the recursion relations which are used to solve the models on the Bethe lattice. The loops in the random graphs thus have no influence on the thermodynamic limit for such ferromagnetic spin models.

We consider the correspondence explicitly for Ising and q state Potts models and also note that multispin interaction models on cacti admit a similar correspondence with a randomized version of the cacti graphs which contain loops.

In this paper, by using our improved plane wave decomposition method, we study the scars in the eigenfunctions of the stadium billiard from a very low state to as high as about the 1 millionth state. In the systematic searching for scars of various types, we have used the approximate criterion based on the quantization of the classical action along the unstable periodic orbit supporting the scar. We have analysed the profile of the integrated probability density along the orbit. We found that the maximal integrated intensity of different types of scars scales in different way with the , which confirms qualitatively and quantitatively the existing theories of scars such as that of Bogomolny and Robnik.

We solve the spectrum of the closed Temperley-Lieb quantum spin chains using the coordinate Bethe ansatz. These models are invariant under the quantum group .

A formalism capable of handling the first step of hierarchical replica symmetry breaking (RSB) in finite-connectivity models is introduced. The emerging order parameter is claimed to be a probability distribution over the space of field distributions (or, equivalently magnetization distributions) inside the cluster of states. The approach is shown to coincide with previous works in the replica-symmetric case and in the two limiting cases m = 0 and 1 where m is Parisi's break point. As an application to the study of optimization problems, the GS properties of the random 3-satisfiability problem are investigated and we present a first RSB solution improving replica-symmetric results.

We investigate analytically the emergence of the categorization ability in the pseudo-inverse attractor neural network. More pointedly, we consider the problem of learning an extensive number of concepts by storing a finite number of examples s of each concept. We find that there is a critical value beyond which the categorization error, as measured by the average fraction of unstable sites in the concepts, decreases monotonically with s.

Non-universal dynamics is shown to occur in a one-dimensional non-equilibrium system of hard-core particles. The stochastic processes included are pair creation and annihilation (with rates and ) and symmetric hopping rates which alternate from one bond to the next (, ). A dynamical scaling relation between the relaxation time and the correlation length in the steady state is derived in a simple way for the case . We find that the dynamical exponent takes the non-universal value .

For the special condition , where the stochastic system is in principle soluble by reduction to a free fermion system, the model is mapped to the Glauber dynamics of an Ising chain with alternating ferromagnetic bonds of values and , in contact with a quantum thermal bath. The full time dependence of the space-dependent magnetization and of the equal time spin-spin correlation function are studied by writing the master equation for this system in the quantum Hamiltonian formalism. In particular, we obtain the dispersion relations and rigorously confirm the results obtained for the correlation length and for the dynamical exponent.

An objective theory of phase shift estimation is formulated within quantum mechanics and quantum estimation theory. The general validity of Bayesian analysis even in the limit of very low particle numbers is demonstrated using the information of measured phase-sensitive data. The formalism was applied to the estimation of an unknown phase in neutron interferometry operating near the quantum limit, when only a few particles are registered, and also in the regime of large particle numbers.

A new operator-based condition for distinguishing classical from nonclassical states of quantized radiation is developed. It exploits the fact that the normal ordering rule of correspondence to go from classical to quantum-dynamical variables does not in general maintain positivity. It is shown that the approach naturally leads to distinguishing several layers of increasing nonclassicality, with more layers as the number of modes increases. A generalization of the notion of sub-Poissonian statistics for two-mode radiation fields is achieved by analysing completely all correlations and fluctuations in quadratic combinations of mode annihilation and creation operators conserving the total photon number. This generalization is nontrivial and intrinsically two-mode as it goes beyond all possible single mode projections of the two-mode field. The nonclassicality of pair-coherent states, squeezed vacuum and squeezed thermal states is analysed and contrasted to one another, comparing the generalized sub-Poissonian statistics with extant signatures of nonclassical behaviour.

We derive a variational characterization for the chemical potential in the Thomas-Fermi model of atoms and molecules. We use this variational principle to obtain accurate upper bounds for the chemical potential as a function of the atomic number Z and the number of electrons N. In particular we study the behaviour of for a weakly ionized atom.

An original mathematical approach for the diffraction of an electromagnetic skew-incident wave by a wedge allows us to reduce such a problem to new elementary functional equations. Using this method, we present an exact analytical solution with closed form expressions for a class of wedge of any angle with a certain type of anisotropic boundary condition which does not require any field component to vanish on the faces. We thereby consider the important case where the principal axes of anisotropy are along and normal to the edge, the relative impedance matrix attached to each face being diagonal with its determinant equal to unity.

We study, in two space dimensions, the heat equation with a random potential that is a white noise in space and time. We introduce a regularization of the noise and prove that, by a suitable renormalization of the coupling coefficient, the covariance has a non-trivial limit when the regularization is removed. The limit is described in terms of a two-body Schrödinger operator with singular interaction.

We study a Hamiltonian coupling a harmonic oscillator to massless scalar bosons which may have arbitrary energy. For certain values of the parameters that this Hamiltonian contains, we prove that the poles of its resolvant matrix elements are not in one-to-one correspondence with the eigenstates of the isolated oscillator. This result raises the question of the validity of this correspondence, even for small coupling, in atom-radiation interaction.

We discuss the use of spectral methods to estimate the decay of correlations of observables in discrete dynamical systems. Some abstract results are stated and applied to the analysis of correlation decay for smooth and analytic observables considered on a particular class of skew-endomorphisms of the 2-torus. General superexponential estimates to correlation decay are also established for the algebraic automorphisms and endomorphisms of the n-torus, whenever hyperbolic or purely expanding conditions occur.

Features of the classical mechanics under the potential are related to the quantum-mechanical spectrum. A known topological obstruction to the existence of global action-angle variables is shown to be reflected in a sharp change in the disposition of eigenvalues around E = 0. The energies of states labelled by a radial quantum number v vary smoothly with the angular momentum for E<0 but have a discontinuous derivative at for E>0. An alternative labelling , where shows the opposite effect. Semiclassical arguments based on the forms of positive and negative energy trajectories also suggest a near symmetry between the positive second derivative ( at energy and the negative counterpart ( at energy , which is confirmed in the quantum-mechanical spectrum. Minor deficiencies in the Bohr-Sommerfeld rule are removed by introducing an analytical phase correction for states with and .

Extended discrete kinetic theory including sources, sinks, creation and annihilation of test particles, inelastic scattering, etc added to the elastic collisions, has recently been introduced, in which, for instance polynomials of the mass are added to the mass conservation law which becomes nonconservative. Here, to the classical two- and three-dimensional Broadwell elastic collision models, we associate nonconservative models adding, to the conservation law, a polynomial of the second degree. We present a method, for the determination of similarity, periodic and (1 + 1)-dimensional exact solutions with one space variable, that we apply to the Broadwell models with Piechor-Platkowski quadratic nonconservative terms.

Using Reshetikhin's construction for multiparametric quantum algebras we obtain the associated multiparametric quantum spin chains. We show that under certain restrictions these models can be mapped to quantum spin chains with twisted boundary conditions. We illustrate how this general formalism applies to construct multiparametric versions of the supersymmetric t-J and U models.

This paper is devoted to a systematic construction of second-order differential equations invariant under the Poincaré, similitude and conformal groups in three-dimensional spacetime. A classification of all possible realizations of the Lie algebras under the action of the group of local diffeomorphisms of is presented. Then by means of the differential invariants the most general invariant differential equations of second order are constructed.

Nonadmissible, weakly admissible and admissible cyclic representations and other algebraic properties of the generalized homographic oscillator (GHO) are studied in detail. For certain ranges of the deformation parameter, it is shown that this new deformed oscillator is a prototype cyclic oscillator endowed with a non-negative (admissible) spectrum. By changing the deformation parameter, the cyclic spectrum can be tuned to have an arbitrarily large period. It is shown that the standard harmonic oscillator is recovered at the nonadmissible infinite-period limit of the GHO. With these properties, the GHO provides a concrete example of an oscillator rich in a variety of cyclic representations. It is well known that such representations are of relevance to the proper algebraic formulation of the quantum-phase operator. Using a general scheme, it is shown that admissible cyclic algebras permit a well-defined Hermitian phase operator of which properties are studied in detail at finite periods as well as at the infinite-period limit. Fujikawa's index approach is applied to admissible cyclic representations and in particular to the phase operator in such algebras. Using the specific example of GHO it is confirmed that the infinite-period limit is distinctively singular. The connection with the Pegg-Barnett phase formalism is established in this singular limit as the period of the cyclic representations tends to infinity. The singular behaviour at this limit identifies the algebraic problems, in a concrete example, emerging in the formulation of a standard quantum harmonic-oscillator phase operator.

The WDVV equations of associativity in two-dimensional topological field theory are completely integrable third-order Monge-Ampère equations which admit bi-Hamiltonian structure. The time variable plays a distinguished role in the discussion of Hamiltonian structure, whereas in the theory of WDVV equations none of the independent variables merits such a distinction. WDVV equations admit very different alternative Hamiltonian structures under different possible choices of the time variable, but all these various Hamiltonian formulations can be brought together in the framework of the covariant theory of symplectic structure. They can be identified as different components of the covariant Witten-Zuckerman symplectic 2-form current density where a variational formulation of the WDVV equation that leads to the Hamiltonian operator through the Dirac bracket is available.

With a new asymptotic expansion for the specific Kummer function that appears in the response theory of low-temperature quantum plasmas in an external magnetic field, the dielectric properties of the charged Bose gas, namely its collective and transverse modes and screening properties, are evaluated in the weak-field limit.

Electronic single-particle states in 1D finite, periodic and quasiperiodic potentials are analysed. The notions of band germs, Bloch germs, bound and decaying states and of energy gauges are introduced and illustrated by exact solutions with the continuous transfer matrix.

We use the maximum-entropy variational technique to infer solutions to second-order partial differential equations. We restrict ourselves to problems with Dirichlet boundary conditions. First, we construct a basis of moment functions in terms of which we set up the constraints for a maximum entropy inversion for the inference of solutions to elliptic equations. Then we extend the scheme to the inference of time dependence in evolution equations. This extension is done by solving a small system of first-order initial-value differential equations for the moments of the solution vector.

Satake superdiagrams corresponding to the Lie superalgebras , , , C(n), together with the exceptional ones are constructed from their Dynkin diagrams with a view to determine the real forms of these superalgebras. The involutive automorphisms are computed from a modified formula to account for all pertinent cases in a consistent way. This mechanism is also used to furnish a general treatment of the Iwasawa and Langlands decompositions of these superalgebras. In particular, we compute these decompositions for for illustrational purposes and obtain the corresponding induced representations using the method of Schmidt construction.

The vibrational levels of diatomic molecules via Morse potentials are studied by means of the system confined in a spherical box of radius . It is shown that there exists a critical radius at which the spectrum of the usual unbounded system can be calculated to any desired accuracy. The results are compared with those of Morse's classical solution which is based on the assumption that the domain of the internuclear distance r includes the unphysical region . By determining numerically exact lower and upper bounds for the energy eigenvalues of molecule, it is deduced here that Morse's approach is perfect and gives very impressive results.

All finite-dimensional indecomposable solvable Lie algebras , having the triangular algebra T(n) as their nilradical, are constructed. The number of non-nilpotent elements f in satisfies and the dimension of the Lie algebra is .

The exact solution to the problem of reflection and diffraction of atomic de Broglie waves by a travelling evanescent wave is found starting with a bare-state formulation. The solution for the wavefunctions, the tunnelling losses and the non-adiabatic losses are given exactly in terms of hyper-Bessel functions, and are valid for all detuning and Rabi frequencies, thus generalizing previous approximate methods. Furthermore, we give the limiting cases of the amplitudes in the uniform semiclassical limit, which is valid in all regions including near the classical turning points, and in the large and weak coupling cases. Exact results for the zero detuning case are obtained in terms of Bessel functions. We find our uniform semiclassical limit to be closer to the exact result over the full range of parameter values than the previously reported calculations.