Table of contents

The general quadratic time-dependent quantum Hamiltonian is analysed using a time-dependent realisation of its SO(2,1) invariance Lie algebra. Several interesting features of this procedure are pointed out. As an example, the authors easily calculate dynamical and geometrical (Berry) phases using only an algebraic procedure. An incorrect result previously published by another author is also pointed out.

The author discusses the characterisation of a class of quasilinear hyperbolic systems which enjoy the property of being separable in two independent equations. A sort of duality relating these with the Lax 'linearly degenerate' systems (1957) is also pointed out.

The authors give a simple but creative closed-form solution of the Fourier transform of a polygonal shape function.

The theory of Hill's equation is applied to periodic orbits in a classical model of the magnetic hydrogen atom. It is shown how an infinite Hill determinant may be approximated and computed, thus giving the discriminant of the relevant Whittaker-Hill equation. This discriminant supplies information on the stability of the above periodic orbits which is much more easily obtained than by the numerical integrations of the orbits appearing in previous publications. The method is also applicable to other classical systems of current interest.

The infinitesimal symmetry transformations associated with the Henon-Heiles problem (1964) are determined. It is shown that the invariance group is a one-parameter group and the corresponding first integral is the energy. The method used indicates that no other symmetry transformations or exact integral exist.

The author introduces a class of one-dimensional tight-binding models with square-well-like potentials, V_n=+or-V, which exhibit a mobility edge in the spectrum at the energy E_c=2-V, provided that V(2. Using the transfer matrix method the author is able to find the analytical expression of both the localisation length and the density of states for the whole range of energies E. The transition from extended to localised states is characterised by a singularity in the spectrum as E_c is approached from below.

The anticommutator analogue of the Baker-Hausdorff lemma is formulated and proved by the differential equation method. It is pointed out that this analogue, when suitably transformed, is more convenient for application whenever the operators in question satisfy simpler repeated anticommutator than repeated commutator relations.

It is shown how the local structure of chaotic repellers, being responsible for transient chaotic behaviour, is deduced from the properties of hyperbolic periodic orbits. Relations between static and dynamical multifractal spectra, with respect to the natural invariant measure on the repeller, are derived for invertible maps of the plane. The results obtained for maps with unit Jacobian apply to Hamiltonian systems with two degrees of freedom which exhibit the phenomenon of irregular scattering and are characterised by an exponential decay of trapping probability.

The authors propose a method to calculate percolation thresholds p_c and their error bars Delta p_c of two-dimensional (2D) lattices. The characteristic feature of their method is an efficient extraction of information from the results of Monte Carlo (MC) simulations. They apply this method to some 2D systems such as a square, Kagome and dice lattice, Penrose tiling and a Penrose dual lattice. Their method enables them to estimate thresholds very accurately even when they use the MC data obtained from fairly small sizes. For instance, they achieve three significant figures for p_c from the MC data obtained from lattices with sizes less than 300*300 and the increment 0.002 of p.

The authors examine whether or not two-dimensional quasi-periodic lattices belong to the same 'universality' class as periodic lattices, and they study bond and site percolation in Penrose tiling and its dual lattice by making use of Monte Carlo simulations. For the sake of comparison, they also investigate percolation in periodic systems such as a square, Kagome and dice lattice. For all these lattices, they evaluate several critical exponents such as alpha related to the total number of clusters, beta related to the percolation strength, gamma related to the mean cluster size, nu related to the correlation length, tau related to cluster sizes, and the fractal dimension D. Their results indicate that universality holds in two-dimensional lattices with or without periodicity where the coordination number could be either single-valued or multi-valued.

The authors examine general learning procedures for neural network associative memories and find algorithms which optimise convergence.

The authors propose a new algorithm for evolution of neural networks, by division of the network into subnets, in order to reach retrieval of stored correlated patterns. They present a fast code for simulations with a large number of neurons ( approximately 10000). The simulations were carried out on a microcomputer.

When constructing series expansions for the zero-field partition function of the q-state Potts model on the square lattice using the finite-lattice method, the series can be extended by one term by adding a correction of (q-1) times the number of convex polygons of an appropriate order. By virtue of duality, such corrections apply for both high- and low-temperature series. For low-temperature series, corrections to the field-dependent partition function are of the same order in the temperature variable as the zero-field correction. The general corrections are given by area-weighted moments of the number of convex polygons. The authors obtain conjectured closed-form expressions for the two generating functions that give corrections to the first two field derivatives of the low-temperature partition function.

The nonlinear wave equations for the complex scalar field invariant under a conformal group are constructed and multiparametrical exact solutions of certain nonlinear complex d'Alembert equations are found.

An exact solution of the spherical Raman-Nath equation (1937) is given. The result is applied to discuss photon statistics and squeezing properties of a free-electron laser.

A set X_Gamma of vector fields in the evolution space E playing the role of Newtonian vector fields, with respect to a second-order equation field Gamma , is introduced and endowed with a C^infinity (E)-module structure. A dual set M^*_Gamma is used for giving an answer to the Lagrangian inverse problem. The symmetry theory is also developed in this framework and, in particular, the characterisation of symmetries of Gamma in terms of the transformation properties of the Lagrangian L is also given.

For a quantum particle moving in the combined field of a negative delta potential and a negative step potential, respectively a semi-infinite Kronig-Penney lattice, the behaviour of a resonance pole is studied as a function of the distance between the centre of the delta potential and the step, respectively lattice. It is found that as this distance is decreased the resonance changes into a virtual state in both cases. In the Kronig-Penney case, however, the situation can become more complicated, particularly if there exists a band gap below zero.

A set of linearly independent functions is orthogonalised by means of the Lowdin transformation (1956). A simple method is presented for calculating the orthogonalisation as a polynomial of finite degree in the overlap matrix S. The method is illustrated for various cases.

Fer's infinite-product expansion (1958) is reconsidered and applied to the specific case of the time-displacement operator in quantum mechanics. An alternative version of this expansion due to Wilcox is also discussed and found to be quite different from the original one. In general the latter is expected to possess better convergence properties.

Fock (1935) explained the degeneracy of the energy levels of the Kepler problem (or hydrogen atom) in terms of the dynamical symmetry group SO(4). He showed that the problem is equivalent to the geodesic flow on the sphere S³. The 'hidden' symmetry SO(4) is made manifest. The classical n-dimensional Kepler problem can be better understood by enlarging the phase space of the geodesical motion on Sⁿ and including time and energy as canonical variables: a following symplectomorphism transforms the motion on Sⁿ in the Kepler problem. The authors prove that the Fock procedure is the implementation at 'quantum' level of the above-mentioned symplectomorphism.

The dynamical evolution is described within the phase-space formalism by means of the Moyal propagator, which is the symbol of the evolution operator. Quadratic Hamiltonians on the phase space are distinguished in that their Moyal bracket with any function equals their Poisson bracket. It is shown that, for general time-independent quadratic Hamiltonians, the Moyal propagators transform covariantly under linear canonical transformations; they are then derived and classified in a fully explicit manner using the theory of Hamiltonian normal forms. The authors present several tables of propagators. It is proved that these propagators belong to the Moyal algebra of distributions, and that the spectrum of the Hamiltonian may be obtained directly as the support of the Fourier transform of the Moyal propagator with respect to time. From that, the quantum-mechanical problem for these Hamiltonians is, in principle, completely solved. The appropriate path-integral formalism for phase-space quantum mechanics, leading back to the same results, is outlined in an appendix.

It is shown that, unlike the model based only on the potential singularity U(x)=- mod x mod ^-1, the group of hidden symmetry explains not only the double degeneration of the energy spectrum but also the explicit form of the spectrum, wavefunctions and an extra constant of motion analogous to the Runge-Lenz vector.

The authors analyse a possible connection between different quantisation schemes and the Bargman-Segal realisation of the Heisenberg algebra H. They show that only a one-parameter subfamily of the family of Heisenberg algebras H_Q subduced from H(+)H can be rewritten in the Bargman-Segal form.

A new algebraic Bethe ansatz construction for the six-vertex model is formulated using intertwining vectors and local gauge transformations. A new set of eigenvectors and their associated eigenvalue equations are derived. This basis is well suited to connect vertex, SOS and RSOS models. Exact relations on a finite lattice between vertex and SOS models are proven with the help of the intertwining vectors.

Non-equilibrium roughening transitions are described and shown to take place as a function of deposition rate and temperature. Discussion is given on the physical mechanism behind this effect, relating its appearance to the relative magnitudes of the vertical and horizontal length scales of surface configurations. The roughening length is defined, and several models are considered, one of which gives a finite transition temperature. Reference to recent experimental results is made and a phase diagram is discussed.

The author considers the energy levels of a generic quantum mechanical system with Hamiltonian H depending on a parameter X. If the energy levels E_n are plotted as a function of X, the curves do not cross. The points of closest approach are called avoided crossings; these have a distinctive geometry and are important because they determine the limits of applicability of the adiabatic theorem. He describes some theoretical results on the parameter space density of avoided crossings, for systems with the spectral statistics of the Gaussian orthogonal ensemble (GOE). These results are in good agreement with numerical experiments.

For pt.I see ibid., vol.22, p.1839, (1989) For the self-avoiding walk problem, the coefficients of the chain generating function and of the generating function for the sum of square end-to-end distances have been extended to 20 terms for the triangular lattice, to 27 terms for the diamond lattice, to 21 terms for the simple cubic lattice and to 16 terms for the BCC lattice. Precise estimates of the critical points are obtained, and for the exponents we find that gamma =1.161+or-0.002 and nu =0.592+or-0.003 encompasses all the three-dimensional lattice data.

Four-loop expansions of the coefficients of the Callan-Symanzik equation for 3D and 2D dilute Ising models are calculated. Fixed-point coordinates and critical exponents are estimated by two methods of summation of double series: a generalisation of the Pade-Borel approximation and the first confluent form of the in algorithm of Wynn. Summation of the double series for the 2D dilute Ising model gives exponents very close to exponents of the pure Ising model, in accordance with the exact solution. The two methods are also applied to summation of single-variable series of pure Ising and polymer models, both in 3D and 2D cases. Both methods are shown to provide close agreement between present estimates and results obtained earlier either exactly in conformal invariant theories (2D) or numerically with high accuracy (3D). An additional test is provided by estimation of critical exponents of the 2D O(n) model for n=-1. The methods tested this way are used to calculate critical exponents of the 3D dilute Ising model. The values obtained are consistent with recent experimental results.

The authors investigate the onset of replica symmetry breaking for the p-state clock spin glasses in a magnetic field. The p=3 case shows a curious behaviour, the instability being of the Gabay-Toulouse type (1981) but with a different exponent; the results are non-invariant under reflection of the field (h to -h). For p=4, the instability is of the Almeida-Thouless type (1978) (Ising-like), despite the fourfold symmetry of the spin variable. For p>or=5, the absence of reflection symmetry on the spin variable is irrelevant for p odd and the behaviour is XY-like.

Spin-glass ordering in conventional models normally reflects, on average, the rotational symmetry of the Hamiltonian. The authors demonstrate that the four-state clock model is exceptional in that the average spin-glass order is essentially collinear (twofold symmetric) despite the fourfold symmetry of the Hamiltonian. Fluctuation effects are predicted only for systems with pure states unrelated by symmetry. However, the XY spin glass with finite fourfold anisotropy is predicted to have conventional (fourfold symmetric/isotropic) ordering.

Given any central (or one-dimensional) potential, the authors discuss the general problem of using supersymmetry transformations to calculate families of potentials which are phase equivalent to it, but have 0, 1, 2, 3, . . . less bound states. Various interesting properties of such families of potentials are derived, and it is shown how the previously studied Abraham-Moses (1980) and Pursey potentials (1986) emerge as limiting cases. For concreteness, the isospectral one-parameter families for two specific potentials of physical interest (the Coulomb and harmonic oscillator potentials) are fully worked out.

Vacuum structure in SQCD is analysed. It is pointed out, in particular, that quantum mechanical tunnelling in the massive theory is mediated by pure instantons and zero classical scalar fields due to the trivial topological structure of the set of minima of the classical potential. In the massless theory, on the other hand, the set of minima has a non-trivial topology, as a result of which an instanton has to be accompanied by a topologically non-trivial scalar field in order to mediate the tunnelling. Indeed, this configuration has an infinite action if a mass term is added, and its tunnelling is completely suppressed unlike that of the trivial scalar configuration.

The authors study the scaling properties of the probability density of random walks on comb-like structures. They find that, in contrast to the first-passage time, in which several exponents are needed to characterise their distribution, the probability density can be described by a single exponent.

An infinite number of forward and reverse n-furcations of tangent bifurcations of unimodal maps can be obtained from a simple substitutional rule. Besides the usual Feigenbaum-type universal constants, new numbers emerge that are independent of the particular map, but for a given n depend on the type of sequence under consideration.

A general formula recently obtained by Cardy (1988), which relates the second moment of the energy-density correlator away from criticality to the value of the conformal anomaly, is verified in the Baxter model (1971). This result is the first analytical confirmation of Cardy's formula in a non-trivial interacting system.