Table of contents

The perturbation approach is used to derive the exact correlation length of the dilute lattice models in regimes 1 and 2 for L odd. In regime 2 the model is the lattice realization of the two-dimensional Ising model in a magnetic field h at . When combined with the singular part of the free energy the result for the model gives the universal amplitude as in precise agreement with the result obtained by Delfino and Mussardo via the form-factor bootstrap approach.

Four results are given that address the existence, ambiguities and construction of a classical R-matrix given a Lax pair. They enable the uniform construction of R-matrices in terms of any generalized inverse of ad L. For generic L a generalized inverse (and indeed the Moore - Penrose inverse) is explicitly constructed. The R-matrices are, in general, momentum dependent and dynamical. The construction applies equally to Lax matrices with spectral parameter.

The integral representation of the orthogonal groups for zonal spherical functions of the symmetric space is used to obtain a generating function for such functions. For the case N = 3 the three-dimensional integral representation reduces to a one-dimensional one.

The quantizations of n identical or distinguishable particles, which are localized on closed orientable 3-manifolds, viewed as a three-fold branched covering of with a matching knot chosen as branching set, are classified up to unitary equivalence. The result connects the set of non-equivalent quantizations to knot theory.

The antiferromagnetic Heisenberg spin chain with N spins has a sector with N= odd, in which the number of excitations is odd. In particular, there is a state with a single one-particle excitation. We exploit this fact to give a simplified derivation of the boundary S matrix for the open antiferromagnetic spin- Heisenberg spin chain with diagonal boundary magnetic fields.

The key equations of the supersymmetric extension of the symplectic Faddeev - Jackiw quantization formalism are written in an alternative way. In this method the crucial problem is to compute the inverse of the symplectic supermatrix. We show how it can be easily given once the configuration space is defined.

By means of Ito calculus it is possible to find, in a straightforward way, the analytical solutions to some equations related to the passive tracer transport problem in a velocity field that obeys the multidimensional Burgers equation and solutions to a simple model of reactive tracer motion.

All Lie symmetries of the Burgers equation driven by an external random force are found. Besides the generalized Galilean transformations, this equation is also invariant under the time reparametrizations. It is shown that the Gaussian distribution of a pumping force is not invariant under the symmetries and breaks them down leading to the non-trivial vacuum (instanton). Integration over the volume of the symmetry groups provides the description of fluctuations around the instanton and leads to an exactly solvable quantum mechanical problem.

We investigate algebraic structure for the -type Calogero model by using the exchange-operator formalism. We show that the set of Jack polynomials whose arguments are Dunkl-type operators provides an orthogonal basis.

The algebraic Bethe ansatz for the integrable vertex model constructed from the four-dimensional representation of the superalgebra gl(2,1) is calculated using a ferromagnetic reference state. This Bethe ansatz was known only for the three-dimensional representation leading to the supersymmetric t-J model. The necessary modification of the nested algebraic Bethe ansatz scheme and generalizations to related models are discussed.

Commuting translations in the phase plane are shown to be observables that are canonically conjugate to the annihilation operator of the harmonic oscillator states. The eigenfunctions and the eigenvalues of these commuting operators form the kq-representation. The eigenfunctions of the annihilation operator are the coherent states. The kq-eigenfunctions and the coherent states are the limiting cases of quantum-mechanical and classical wavefunctions, respectively, and it is satisfying to discover that they relate to canonically conjugate operators.

An open quantum model of a small system strongly interacting with a thermodynamic bath is suggested which binds cyclically, upon working in isothermal conditions, particles (atoms, molecules or molecular groups) to bound states. The latter may be even energetically disadvantageous. The effect is due to relaxation processes in intermediate states upon combined scattering of the particles on the system and can be viewed as an induced self-organization described here by a linear theory.

We consider the single-spin-flip dynamics of the random-field Ising model on a Bethe lattice at zero temperature in the presence of a uniform external field. We determine the average magnetization as the external field is varied from to by setting up the self-consistent field equations, which we show are exact in this case. The qualitative behaviour of magnetization as a function of the external field unexpectedly depends on the coordination number z of the Bethe lattice. For z = 3, with a Gaussian distribution of the quenched random fields, we find no jump in magnetization for any non-zero strength of disorder. For , for weak disorder the magnetization shows a jump discontinuity as a function of the external uniform field, which disappears for a larger variance of the quenched field. We determine exactly the critical point separating smooth hysteresis curves from those with a jump. We have checked our results by Monte Carlo simulations of the model on three- and four-coordinated random graphs, which for large system sizes give the same results as on the Bethe lattice, but avoid surface effects altogether.

We reconsider the quantum inverse scattering approach to the one-dimensional Hubbard model and work out some of its basic features so far omitted in the literature. It is our aim to show that the R-matrix and monodromy matrix of the Hubbard model, which have now been known for ten years, have good elementary properties. We provide a meromorphic parametrization of the transfer matrix in terms of elliptic functions. We identify the momentum operator for lattice fermions in the expansion of the transfer matrix with respect to the spectral parameter and thereby show the locality and translational invariance of all higher conserved quantities. We work out the transformation properties of the monodromy matrix under the su(2) Lie algebra of rotations and under the -pairing su(2) Lie algebra. Our results imply invariance of the transfer matrix for the model on a chain with an even number of sites.

Starting from Langevin equations, we derive Fokker - Planck-like equations (FPLEs) for the joint distribution of displacements and velocities, p(x,v,t), for a particle in a Gaussian random force field, firstly for the inertial process (i.e. in the absence of a frictional force) with a time correlated force, and secondly, for the Brownian motion with a white-noise force. From two different forms of the Langevin equation as coupled or decoupled first-order equations, we obtain two different forms of FPLEs for each one of these processes. In the inertial case one of the FPLEs reduces to an equation studied earlier by the author, while the other coincides with the equation obtained recently by Drory from an involved time discretization. In the Brownian motion case one of the FPLEs coincides with the free-particle Kramers equation obtained from the Fokker - Planck formalism for Markov processes. For each one of these processes the exactly determined initial value solutions of the two FPLEs are found to coincide. It follows, in particular, that the Markovian character of p(x,v,t) for the Brownian motion is respected, regardless of which FPLE is used for defining it. Furthermore, for each process the two FPLEs lead to the same diffusion-like equation for the marginal distribution of displacements. The latter have been used elsewhere for studying first passage times, as well as survival probabilities in the presence of traps.

In this paper we begin by giving a description of functional methods of quantum field theory for systems of interacting q-particles. These particles obey exotic statistics and are the q-generalization of the coloured particles which appear in many problems of condensed matter physics, magnetism and quantum optics. Motivated by the general ideas of standard field theory we prove the q-functional analogues of Hori's formulation of Wick's theorems for the different ordered q-particle creation and annihilation operators. The formulae have the same formal expressions as fermionic and bosonic ones but differ by the nature of fields. This allows us to derive the perturbation series for the theory and develop analogues of standard quantum field theory constructions in q-functional form.

We study chaos in a two-dimensional Ising spin glass by finite temperature Monte Carlo simulations. We are able to detect chaos with respect to temperature changes as well as chaos with respect to changing the bonds, and find that the chaos exponents for these two cases are equal. Our value for the exponent appears to be consistent with that obtained in studies at zero temperature.

Quasiperiodic, planar Ising models with ferromagnetic nearest-neighbour interactions should show the same universal critical behaviour as the classical Ising model on the square lattice. We use the eightfold symmetric Ammann - Beenker tiling to investigate this and employ the distribution of the Lee - Yang and the temperature zeros of the partition function in the complex plane. Our results support, as expected, the existence of an Onsager-type phase transition, i.e. a second-order transition with critical exponents , and .

By applying Lieb's spin-reflection-positivity method and exploiting a commutation relation satisfied by the negative-U Hubbard Hamiltonian, we prove two rigorous theorems on the binding energy of fermions, the one-particle and the two-particle gaps for the model on an arbitrary finite lattice.

The Green function of a square tight-binding model in magnetic fields is expressed by means of continued fractions. For rational fields, the expression is reduced to a more useful one. The Green function expressing the out-going wave is numerically computed at arbitrary sites for the first time. It is found that the absolute value of the Green function takes maximum values on the reciprocal lattice of the magnetic Brillouin zone.

The hull-gradient method is used to determine the critical threshold for bond percolation on the two-dimensional Kagomé lattice (and its dual, the dice lattice). For this system, the hull walk is represented as a self-avoiding trail, or mirror-model trajectory, on the (3,4,6,4)-Archimedean tiling lattice. The result (one standard deviation of error) is not consistent with previously conjectured values.

We prove that the probability of finding a scattered quantum-mechanical particle at large times in a truncated cone is identical with the scattered flux, integrated over time, across a distant spherical surface subtending this cone. The theory applies to potentials with arbitrary local singularities and decaying faster than at large distances.

We find a new class of time-dependent partial waves which are solutions of the time-dependent Schrödinger equation for three-dimensional harmonic oscillator. We also show the decomposition of coherent states of harmonic oscillator into these partial waves. This decomposition appears to be particularly convenient for a description of the dynamics of a wavepacket representing a particle with spin when the spin - orbit interaction is present in the Hamiltonian. An example of an evolution of a localized wavepacket into a torus and backwards, for particular initial conditions is analysed in analytical terms and shown with computer graphics.

We consider the problem of evaluating the Casimir effect by the mode-sum method for a quantum field in a one-dimensional space, in the presence of two point-like boundaries of Dirichlet type, and under the influence of a constant external field - which may be envisaged as a classical gravitatory field near the surface of a planet or a classical electric field in the interior of a flat capacitor. The case of infinitely separated points is also examined. Despite apparent simplicity, the calculation exhibits rather non-trivial aspects. The possibility of an experimental observation of these effects is considered.

After obtaining some useful identities, we prove an additional functional relation for q-exponentials with reversed order of multiplication, as well as the well known direct one, in a completely rigorous manner.

A complete choice of generators of the centre of the enveloping algebras of real quasisimple Lie algebras of orthogonal type, for arbitrary dimension, is obtained in a unified setting. The results simultaneously include the well known polynomial invariants of the pseudo-orthogonal algebras , as well as the Casimirs for many non-simple algebras such as the inhomogeneous , the Newton - Hooke and Galilei type, etc, which are obtained by contraction(s) starting from the simple algebras . The dimension of the centre of the enveloping algebra of a quasisimple orthogonal algebra turns out to be the same as for the simple algebras from which they come by contraction. The structure of the higher-order invariants is given in a convenient `pyramidal' manner, in terms of certain sets of `Pauli - Lubanski' elements in the enveloping algebras. As an example showing this approach at work, the scheme is applied to recovering the Casimirs for the (3 + 1)-kinematical algebras. Some prospects on the relevance of these results for the study of expansions are also given.

The notion of a co-isotropic and Legendre - Lagrangian submanifold of a Jacobi manifold is given. A characterization of conformal Jacobi morphisms and conformal Jacobi infinitesimal transformations is obtained as co-isotropic and Legendre - Lagrangian submanifolds of Jacobi manifolds.

Maps for nonlinear evolution equations are discussed. An algorithmic method for deriving rational solutions is presented. This approach is illustrated by examples which have solutions in the form of two truncated expansions.

A finite expansion of the exponential map for a matrix is presented. The method uses the Cayley - Hamilton theorem for writing the higher matrix powers in terms of those for the first N - 1. The resulting sums over the corresponding coefficients are rational functions of the eigenvalues of the matrix.

We use simple diagrammatic techniques to analyse the ordinary representation theory of the Hecke algebras , and to construct modules (resp. representations) which are generically simple (irreducible) and well defined in every specialization of q, including roots of unity. We determine several physically important properties of these modules, generalizing properties of the Temperley - Lieb algebra and its diagrams which have proved useful for lattice models.

We show how these results can be used to locate energy level crossings in invariant quantum spin chains, and locate a new crossing of the thermodynamic limit spin chain at as an example.

The case of an electron tunnelling through a parabolic repeller and acted upon by a microwave electric field is presently treated taking account of dissipation. Dissipation is introduced using a frictional force and the relevant quantum mechanics stems from an appropriate Lagrangian capable of generating the dissipative force. Considerations are presented for deriving a suitable continuity equation adapted to the dissipative processes involved. Starting with a wavepacket as the particle's initial state, expressions for the probability and current densities, as well as the transmission coefficient, are derived. These are used to see the influence of friction, frequency, amplitude and initial phase of the applied field on the tunnelling effect. Furthermore, the localizing effect of friction is made visible on the reflected portion of the wavepacket, where the main body of the probability resides.

K-theory allows us to define an analytical condition for the existence of `false' gauge field copies through the use of the Atiyah - Singer index theorem. After establishing this result we discuss a possible extension of the same result without the help of the index theorem and suggest possible related lines of work.

In relativistic Schrödinger theory, additional conservation laws arise of topological origin. These are due to the existence of topological currents which are built up by the exclusive use of operators, whereas the matter currents are composed of the densities. The general concepts and results are exemplified by considering a specific (Dirac) spinor field over the Robertson - Walker universes. The invariant, associated to the topological current, can be explicitly determined for -bundles.

The poles of the S-matrix for a two-channel model with square well potentials are calculated. It is found that the trajectories of these poles in the complex k-plane for varying coupling strength show avoided crossings. At the critical parameters the poles coincide to form a double pole. Its presence is revealed by a symmetry property of the cross section.

The phase space lattice Hamiltonian is a realistic model for Bloch electrons in a magnetic field. It has a fractal spectrum when the lattice has centres of threefold or fourfold rotational symmetry. This fact has been explained using a renormalization group (RG) method, assuming that the RG transformation preserves the symmetry of the Hamiltonian. The symmetry preservation property has previously been demonstrated for fourfold rotation; the threefold case is considerably more difficult to analyse. In this paper we present a simplified form of the RG equations which clearly exhibits the threefold symmetry preservation. We also discuss the case of sixfold rotational symmetry, for which the symmetry of the Hamiltonian may be reduced to threefold under the action of the RG.

This paper is based on Nigmatullin's study. When the `residual' memory set is a generalized cookie-cutter set on , using various hypotheses it is proved that the fractional exponent of a fractional integral is not uniquely determined by the fractal dimension of the generalized cookie-cutter set. It is determined by of self-similar measure (or infinite self-similar measure) on this generalized cookie-cutter set, and can run over all positive real numbers.

The mathematical analysis in 1996 J. Phys. A: Math. Gen. 29 1767, is not sufficient to decide whether in one dimension the singularities of the potentials and split the corresponding one-particle quantum systems at the origin into two completely decoupled subsystems. In fact, it is argued that this question cannot be answered by mathematical considerations alone.

The differential operator , , in one dimension is studied using distribution theory. It is proven that there exists a unique self-adjoint operator corresponding to the differential expression understood in the principle-value sense. Point interactions determined by the singular operator are studied.

The following table was omitted from the original article: Submatrix .

Here

Equation (12) was incorrectly printed in this comment. The correct version is: