Table of contents

In contrast to a stationary Gaussian random function of a real variable which is free to have any correlation function, the closest analogous analytic random function in the complex plane has no true freedom - it is (statistically) unique. Since it has arisen only recently, as an apparently universal feature in the physical context of quantum chaos, I refer to it here as `the chaotic analytic function'. I note that it is implied by the assumption that a quantum chaotic wavefunction has Gaussian randomness and has a constant value for the average of its Wigner function in phase space. Interpreted literally this shows that the chaotic analytic function is the Bargmann function of a pure `white noise' wavefunction. More physically, if `constant' is replaced by `smooth on the scale of a Planck area', these assumptions are the semiclassical ones made by Berry for chaotic eigenstates. The analysis shows that the chaotic analytic function is still obtained semiclassically.

Different characteristics of an output signal (average position, population, average energy) are calculated for a particle moving in a piecewise potential and subject to external periodic and random forces. Particular emphasis has been placed on the dependence of these characteristics on the strength of the noise and the frequency of an external field. An external periodic force strives to equalize the populations of the discrete levels, or even to reverse the populations for the space-extended systems. The populations of the potential wells in a one-barrier system subject to oscillation of the wells can either decrease or increase (compared with the field-free case) depending on the frequencies of the external field. All these changes of populations induced by an external field have some resemblance to the similar quantum mechanical problems.

We investigate the effect of the jamming transition on short-range correlation functions in the Nagel-Schreckenberg cellular automaton model of single-lane traffic. At high densities the structure of the time-dependent correlation functions is double peaked. One peak corresponds to moving cars, the other to blocked cars. The emergence of the latter peak as well as the occurrence of short-range anticorrelations behind the propagating peak is related to the jamming transition. At even higher densities the peak corresponding to moving cars disappears which is an indication of a superjamming transition.

For the two-dimensional kinetic Ising model at finite temperature, the local mean magnetization , simply related to the fraction of time spent by a given spin in the positive direction, has a limiting distribution, singular at , the Onsager spontaneous magnetization. The exponent of this singularity defines the persistence exponent . We also study first passage exponents associated to persistent large deviations of , and their temperature dependence.

We consider Ising models with long-range ferromagnetic pair interactions decaying as for . We first find approximate values for the critical temperature. We use a cluster mean-field approach combined with finite-size scaling and Vanden Broeck and Schwartz transformations. For we find which can be compared with recent results of Luijten and Blöte who found , and which is two orders of magnitude more accurate than any previous results. Since we use a mean-field cluster approximation as part of our approach, the accuracy for larger values of decreases significantly. In addition to we obtain approximate values for the critical exponents , and using the coherent anomaly method. For we obtain , , and - all extremely close to the predictions of renormalization group calculations which say that these exponents should take on their classical values for this value of .

We construct the exact ground state for an antiferromagnetic spin- model on the two-leg ladder as an optimum ground state. The ground state contains a discrete parameter and a continuous parameter a which controls z-axis anisotropy. For most values of a the global ground state is unique. It has vanishing sublattice magnetization and exponentially decaying correlation functions. By using the transfer matrix technique, we calculate exactly the fluctuations of the magnetization, the nearest-neighbour correlation, and the longitudinal correlation length as functions of the parameters.

It is demonstrated that the 60 rays corresponding to antipodal pairs of vertices of the 600-cell, and the 300 rays corresponding to antipodal pairs of vertices of the 120-cell, can both be used to give noncolouring proofs of the Bell-Kochen-Specker theorem in four dimensions.

The problem of photon creation from vacuum due to the non-stationary Casimir effect in an ideal one-dimensional Fabry-Perot cavity with vibrating walls is solved in the resonance case, when the frequency of vibrations is close to the frequency of some unperturbed electromagnetic mode: , , ( is the mean distance between the walls). An explicit analytical expression for the total energy in all the modes shows an exponential growth if is less than the dimensionless amplitude of vibrations , the increment being proportional to . The rate of photon generation from vacuum in the th mode goes asymptotically to a constant value , the numbers of photons in the modes with indices being the integrals of motion. The total number of photons in all the modes is proportional to in the short-time and in the long-time limits. In the case of strong detuning the total energy and the total number of photons generated from vacuum oscillate with the amplitudes decreasing as for . The special cases of p = 1 and p = 2 are studied in detail.

Making use of an ansatz for the eigenfunctions, we obtain an exact closed-form solution to the non-relativistic Schrödinger equation with the anharmonic potential, in two dimensions, where the parameters of the potential, a, b and c satisfy some constraints.

Nonholonomic systems with an invariant measure: the Suslov, Chaplygin and Veselov-Veselova problem are considered. New families of integrable potential perturbations of these non-Hamiltonian systems are constructed. We also obtain a similar result for a classical Hamiltonian system of the motion of a rigid body fixed at a point.

We give an explicit construction of the coherent states for an arbitrary irreducible representation. We also construct the symplectic structure on the manifold of coherent states, find canonical variables and discuss various classical limits of quantum-mechanical systems with relevant observables that obey commutation relations.

The space of polynomials maps onto itself under affine transformations, . This suggests that a moment reformulation of continuous wavelet transform (CWT) theory (the affine convolution, , of a signal, or wavefunction, ) should lead to significant simplifications in its implementation. We present a comprehensive formalism, with numerical examples, that inextricably links moment quantization (MQ) and CWT theory. For rational fraction potential problems and mother wavelets of the form (Q(x) an appropriate polynomial), MQ permits a more efficient and accurate (in a pointwise convergent sense) CWT implementation; whereas, CWT broadens the scope of applicability for MQ methods, and is its natural extension when a more global approximation is desired. Our formalism also gives one justification for the empirical superiority manifested by previous MQ studies, as compared with dyadic wavelet reconstruction methods. We implement our formalism in the context of the quartic, sextic and octic anharmonic oscillator potentials, and demonstrate the flexibility of the method by treating both the Mexican hat wavelet transform, as well as that based on the mother wavelet .

The massive N-flavour Schwinger model is analysed by the bosonization method. The problem is reduced to the quantum mechanics of N degrees of freedom in which the potential needs to be self-consistently determined by its ground-state wavefunction and spectrum with given values of the parameter, fermion masses, and temperature. Boson masses and fermion chiral condensates are evaluated. In the N = 1 model the anomalous behaviour is found at and . In the N = 3 model an asymmetry in fermion masses removes the singularity at and T = 0. The chiral condensates at are insensitive to the asymmetry in fermion masses, but are significantly sensitive at . The resultant picture is similar to that obtained in QCD by the chiral Lagrangian method.

Forms of the Poisson summation formula (PSF) appropriate for the summation of semi-infinite and infinite Fourier series are derived. Application of these results to the acceleration of convergence of various types of series with monotonically decreasing coefficient functions yields transformed series with terms that decay either exponentially or with the inverse first or second power of the index variable. These two very different convergence properties are explained in terms of the asymptotic properties of the relevant Fourier transforms, which are in turn related to the power series expansions of the summand functions in the original Fourier series. The result is that the Poisson summation formula works best for Fourier cosine series in which the summand functions are expansible in even powers, and for Fourier sine series in which the summand functions have power series with odd powers. Here, application of the PSF produces series of terms that decay exponentially with increasing argument x. In contrast, application of the semi-infinite version of the PSF to Fourier cosine series of terms with odd-power expansions, or to Fourier sine series of terms with even-power expansions yields transformed series involving functions of the form , which decay approximately as . If the summand function in the Fourier series has a power series with both even and odd powers, the transformed series involves sine and cosine integral functions, which decay approximately as . Fourier series of these last three types in general require additional acceleration, for example, by application of the Kummer transformation.

For a periodic Hamiltonian, periodic dynamical invariants may be used to obtain non-degenerate cyclic states. This observation is generalized to the degenerate cyclic states, and the relation between the periodic dynamical invariants and the Floquet decompositions of the time-evolution operator is elucidated. In particular, a necessary condition for the occurrence of cyclic non-adiabatic non-Abelian geometrical phase is derived. Degenerate cyclic states are obtained for a magnetic dipole interacting with a precessing magnetic field.

We compute the finite-size spectrum for the spin- chain with twisted boundary conditions, for anisotropy in the regime , and arbitrary twist . The string hypothesis is employed for treating complex excitations. The Bethe ansatz equtions are solved within a coupled nonlinear integral equation approach, with one equation for each type of string. The root-of-unity quantum group invariant periodic chain reduces to the chain with a set of twist boundary conditions (, an integer multiple of ). For this model, the restricted Hilbert space corresponds to a unitary conformal field theory, and we recover all primary states in the Kac table in terms of states with specific twist and strings.

It has been universally assumed that the spectrum of the magnetohydrodynamics equations, linearized around an equilibrium state, provides enough information on the short-term evolution of the plasma to study certain stability properties. We show that this is true if one takes into account viscous and resistive effects and the equilibrium satisfies certain regularity conditions.

We propose a method of finding approximate solutions associated with nonlinear partial differential equations involving a small parameter by using equivalence transformations. As an application of our method we consider a one-dimensional nonlinear heat conduction model and study its approximate solutions.

In previous papers, with the same series title, an ab-initio procedure was developed for deriving a Lorentz invariant equation with arbitrary spins. This equation is linear in the four momentum , and its coefficients are matrices that can be expressed in terms of ordinary spin and what we called sign spin. In the present paper we consider this equation in an external field which implies just replacing by and discuss the cases when (, being the frequency of the oscillator), and , corresponding respectively to harmonic oscillator potential and a constant magnetic field. By using an appropriate complete set of states, with part of them characterized by the irreps of the chain of groups where the subscripts s and t respectively stand for the ordinary and sign spin, the problem can be formulated in a matrix representation whose diagonalization gives the energy spectrum. For simplicity we shall only consider the symmetric representation of for which s = t, and our interest is focussed on the case when the external field is weak, which gives the non-relativistic limit, and where a perturbation analysis can be applied. We show that the expected non-relativistic result can be obtained only when the sign spin projection takes its maximum value, i.e. when all individual states contributing to the final one correspond to positive energies. In the case of constant magnetic field, we obtain the gyromagnetic ratio consistent with other derivations.

We derive a semiclassical quantization for a spin, study it for not too small a spin quantum number , and compute the 2S+1 eigenvalues of a Hamiltonian exhibiting resonant tunnelling as the magnetic field parallel to the anisotropy axis is increased. Special attention is paid to the resonance condition. As a corollary we prove that semiclassical quantization and quantum-mechanical perturbation theory agree there where they should.