Table of contents

A most general algebra with three generators and with ladder and duality properties, GAW(3), is considered. It is shown that the obtained algebra corresponds to (p,q)-deformation of the Askey-Wilson algebra. For the new algebra GAW(3) the ladder representation is constructed and the overlap problem is discussed.

We present an exact analysis of learning a rule by on-line gradient descent in a two-layered neural network with adjustable hidden-to-output weights (backpropagation of error). Results are compared with the training of networks having the same architecture but fixed weights in the second layer.

Optimal paths in disordered systems are studied using two different models interpolating between weak and infinitely strong disorder. In one case, exact numerical methods are used to study the optimal path in a two-dimensional square lattice whereas a renormalization-group analysis is employed on hierarchical lattices in the other. The scaling behaviour is monitored as a function of parameters that tune the strength of the disorder. Two distinct scenarios are provided by the models: in the first, fractal behaviour occurs abruptly as soon as the disorder widens, while in the other it emerges as a limiting case of a self-affine regime.

Two fundamental quantum signatures of classically chaotic behaviour, large second derivatives and the logarithmic time barrier separating classical from quantum dynamics, have been observed near the separatrix of an integrable spin system. A systematic study in the neighbourhood of the separatrix reveals that the error in semiclassical eigenvalues (measured in units of the quantum level spacing) approaches a finite value at the classical limit. The generic factor which limits the validity of the correspondence principle is the presence of classical instabilities; chaotic behaviour is one such case; the separatrix of an integrable system is another.

Explicit analytic calculations concerning transport coefficients are presented for a relativistic magnetized and collisional plasma, especially at moderately-relativistic (10 keV<or=kT<or=1 GeV) and ultra-relativistic (kT>1 GeV) temperature limits covering up to Planck's scale. Transport coefficients, such as electrical resistivity ( eta _R) and thermal conductivity ( kappa _R) except the thermoelectric coefficient ( lambda _R), however, strikingly decrease with increase in thermal energy ( kappa T), particularly in a moderately-relativistic temperature limit, whereas in an ultra-relativistic temperature regime the trend is similar, although it is rather more drastic. Our results are of importance for understanding collisional transport processes occurring in astrophysical situations and laboratory diagnostic experiments.

The relaxation of the Sherrington-Kirkpatrick model for spin glasses is studied at low temperatures. A recently developed numerical method is used by which an infinitely large system is simulated, hence finite size effects are avoided. For parallel dynamics the decay of the energy as well as of the magnetization is investigated. For low temperatures we find evidence for a relaxation into a state which is characterized by non-equilibrium values of the energy and a non-vanishing remanent magnetization.

The contribution of thermal electron-positron pairs to the thermodynamic properties of black-body radiation (BBR) is considered. This contribution is examined in the vicinity of the temperature corresponding to the electron rest mass energy T_c=m_cc²/ kappa , in which it becomes appreciable. The correction factors are defined as the ratio of extended thermodynamic expressions of BBR (which also include the contributions from thermal pairs) to the familiar expressions of it. Then the variation of these factors with temperature is given. It is shown that while they have the same value for each property in both high and low temperature regimes (T>>T_c and T<<T_c), they have different values about 0.5 T_c. It is found that for BBR, the ratio of the energy density to the pressure has a maximum about 0.33 T_c and also the ratio of the specific heats has a minimum about 0.4 T_c.

The local distortion of a quantum scalar field phi by isolated point and (straight-) line boundaries is studied using simple image-charge constructions. Non-zero temperature is considered. The distortion (away from uniformity) of the phi virtual-particle sea and thermal gas by these boundaries is calculated exactly. Periodic arrays of point and line boundaries are as easily dealt with.

Two kinds of coherent states are constructed in the context of the Calogero-Sutherland singular oscillator. The motion of the peaks of the wavefunctions of these coherent states are compared with the classical trajectory. It is found that while the wavefunction for one kind of coherent states is always singly peaked, that for the other acquires multiple peaks close to the classical turning point near the origin. The two coherent states are found to exhibit a kind of complementarity.

We extend an earlier investigation of a class of m-dimensional lattice sums which were phase modulated by a continuous twist parameter tau =( tau _l... tau _m) and involved Bessel functions J_nu (x), with argument x=2 upsilon q (q identical to mod q mod = square root (q_l²+...+q_m²);q_j=integer For all j in m) such that 0<or= upsilon < pi tau . This investigation-motivated partly by the Henkel-Weston conjectures and partly by the propositions of Ortner and Wagner on the solution of the hyperbolic differential operators-generalizes Previous analyses to include an m-dimensional shift parameter a and an arbitrary scalar b, so that x is now equal to 2 upsilon square root ( mod q+a mod ²+b²); at the same time, it extends the range of upsilon from 0 to infinity . While these generalizations are broad enough to make this a worthwhile study in its own right, the main interest here lies in applying this special technique to such diverse physical problems as finite-sized ferromagnetic or quantum fluid model systems undergoing phase transitions, the Casimir effect and topological mass generation, etc. Consequently, there arise important connections to the zeta function regularization and to heat kernel techniques.

The usual second harmonic generation effective Hamiltonian is shown to be equivalent to a one-dimensional Schrodinger operator with a sextic polynomial potential. This operator belongs to the class of quasi-exactly solvable models, for which a finite number of eigenvalues and eigenvectors can be determined exactly. A Jeffreys-Wentzel-Kramers-Brillouin analysis provides accurate asymptotic expansions for these eigenvalues in the limit of large unperturbed energies.

Perturbations of WD_n and W₃ conformal theories which generalize the (1,2) perturbations of conformal minimal models are shown to be integrable by 'counting arguments'. The A_2n-l,q² and D_4,q³ of corresponding S-matrices are conjectured and proved by explicit construction of conserved non-local charges in the WD₃ case with the proper quantum symmetry group. The gradation change of the known R-matrix with A₅² symmetry from homogeneous to spin, which turns out to be relevant for the perturbation considered, is shown to make the R-matrix crossing invariant and fixes the effective coupling constant as a function of the initial one. Using these results the fundamental S-matrix for vector perturbed minimal WD₃ models is constructed on the basis of the RSOS model with the corresponding symmetry. The S-matrix is checked to reproduce the known results in two particular cases of minimal model number: k=l and k to infinity .

Using the complex WKB method, new semiclassical spectral series for perturbed states of an 'anisotropic' hydrogen atom, taking into account the electron's spin polarization in a homogeneous magnetic held (the anomalous Zeeman effect) are constructed. They are valid uniformly for all values of the field's magnitude. These series correspond to families (stable in the linear approximation) of periodic and conditionally periodic motions of a classical electron in the plane orthogonal to the direction of the field. The corresponding semiclassical wavefunctions have the property of quantum 'superscaring' in the coordinate space near these classical trajectories of motion. We obtain the limit cases of our semiclassical energy levels for strong and weak magnetic fields. We compare them with the only known approximations, i.e. for the hydrogen atom.

For pt.I, see ibid, p.5799-5810. On the basis of the complex WKB method, new spectral series for perturbed states of an 'anisotropic' hydrogen atom, taking into account the electron's spin polarization in a homogeneous magnetic field are constructed. These series correspond to conditionally periodic motions of a classical electron in the plane orthogonal to the field direction. The semiclassical wavefunctions are regular in the entire configuration space including the focal points and have the 'superscar' properties near the domain of classical motion, i.e. near the projection of stable two-dimensional Lagrangian tori on the configuration space. Both for Schrodinger and Pauli operators the analytical description of the semiclassical wavefunctions 'superscar' structure is obtained. This structure is non-uniform with respect to the space coordinates and has singularities with respect to the parameter h(cross),h(cross) to 0, near focal points. In the limit case of strong magnetic fields explicit analytical formulae for the semiclassical energy levels are obtained.

We extend the concept of conjugate vertex operators, first introduced by Dotsenko (1990) in the case of the bosonization of the SU(2) conformal field theory, to the bosonization of the dynamical vertex operators (type II in the classification of the Kyoto school) of the higher spin XXZ model. We show that the introduction of the conjugate vertex operators leads to simpler expressions for the N-point matrix elements of the dynamical vertex operators, that is, without redundant Jackson integrals that arise from the insertion of screening charges. In particular, the two-point matrix element can be represented without any integral.

Generalized KdV hierarchies associated by Drinfeld-Sokolov reduction with grade 1 regular semisimple elements from non-equivalent Heisenberg subalgebras of a loop algebra G(X)C( lambda , lambda ^-1) are studied. The graded Heisenberg subalgebras containing such elements are labelled by the regular conjugacy classes in the Weyl group W(G) of the simple Lie algebra G. A representative w epsilon W(G) of a regular conjugacy class can be lifted to an inner automorphism of G given by w=exp (2i pi ad I₀/m), where I₀ is the defining vector of an sl₂ subalgebra of G. The grading is then defined by the operator d(m,I₀)=m lambda (d/d lambda )+ad I₀ and any grade 1 regular element Lambda from the Heisenberg subalgebra associated with (w) takes the form Lambda =(C₊+ lambda C_-), where (I₀, C_-)=-(m-1)C_- and C₊ is included in an sl₂ subalgebra containing I₀. The largest eigenvalue of adI₀ is (m-1) except for some cases in F₄, E_6,7,8. We explain how these Lie algebraic results follow from known results and apply them to construct integrable systems. If the largest ad I₀ eigenvalue is (m-1), then using any grade 1 regular element from the Heisenberg subalgebra associated with (w) we can construct a KdV system possessing the standard W-algebra defined by I₀ as its second Poisson bracket algebra. For G a classical Lie algebra, we derive pseudo-differential Lax operators for those non-principal KdV systems that can be obtained as discrete reductions of KdV systems related to gl_n. Non-Abelian Toda systems are also considered.

Electrically charged scalar particle-antiparticle pair creation in the field of a magnetic monopole is studied by path integrals. It is found that a magnetic charge g creates pairs with charge e, with angular momentum l<eg- 1/2 with the azimuthal component m=0.

Some specific conditionally exactly solvable potentials are discussed within the path integral formalism. They generalize the usually known potentials by the incorporation of fractional power behaviour and strongly anharmonic terms. We find four different kinds of such potential: the first is related to the Coulomb potential, the second is an anharmonic confinement potential, and the third and fourth are related to the Manning-Rosen potential.

We examine the characteristics of quantum jumps in a Raman ( Lambda ) system driven by two coherent fields in terms of the joint probability for successive arrivals of photons. In particular, we derive the general expression for P(1, T₁; 0, T₂; 1, T₃), the probability that a photon is counted in the interval (t,t+T₁) and (t+T₁+T₂, t+T₁+T₂+T₃) with no photon in the intermediate interval (t+T₁, t+T₁+T₂) and discuss its behaviour.

Using a first-order Casimir operator calculated in a non-standard realization for the so(3,1) algebra, we obtain a one-dimensional scattering problem with LS-type interaction terms. It is shown that for this realization the square of this operator can be expressed in terms of the usual quadratic Casimir. Due to this constraint the scattering states are completely specified by restricting the possible set of eigenvalues accordingly. The results show that the use of extra Casimir operators can provide additional insight into the group theoretical structure of the scattering problem. A generalization for the so(2n-1,1), n>2 case is also given. The underlying supersymmetry of the resulting Schrodinger equations is pointed out. The supersymmetric charge operators are related to our first-order Casimir operators.

Starting from the phase-space generating functional of the Green's function for a constrained Hamiltonian system, the canonical Ward identities under the global symmetry transformation in phase space is deduced. The local transformation connected with this global symmetry transformation is studied, the conserved charges are obtained at quantum level if the effective canonical action is symmetric (the constraints are also invariant under the transformation) and, therefore, the canonical Noether theorem in the quantum case is obtained. The generalized canonical Ward identities under the local transformation has been deduced. We give a preliminary application to a system of an interacting polaron with a photon. The conserved charges and Ward identities for proper vertices are obtained, but we do not carry out the integration over the canonical momenta in the phase-space generating functional as usually performed.

Using the formal series symmetry approach, we obtain three sets of generalized symmetries of the AKNS-type breaking soliton equation. These sets of formal series symmetries are not truncated except that the arbitrary functions of the formal series symmetries are fixed as polynomials of the corresponding independent variables. Differing from the truncated cases such as KP and Toda field equations, these non-truncated symmetries constitute another type of generalization of the Virasoro algebra.

We consider the s-state Potts model on the diamond hierarchical lattice for large s. We show that the generalized dimensions D_q of the density of zeros supported by the associated Julia set are given by D_q = 1 - (q - 1)|s|^-2/3/4 log 2+O(s^-1). The information dimension D₁ equals 1 to all orders.

A single-field bilinear system generating the so-called non-local Boussinesq equation is constructed. From the bilinearization procedure it can be seen that the associated hierarchy of soliton systems which we construct shares part of the solution set of a hierarchy related to the Kadomtsev-Petviashvili equation. A bilinear representation of the recursion operator for the Kaup hierarchy is essential in the construction and a systematic way of obtaining such a representation from just two-soliton considerations is presented.