Table of contents

The connection between quantum mechanical systems with broken and unbroken supersymmetry is discussed. The class of superpotentials is given for which it is possible to find eigenvalues and eigenfunctions by recovering the broken supersymmetry.

The authors introduce a scale-dependent multifractal analysis which is applicable to any data including non-fractal fluctuations. It is found that a white noise can be characterized by a scale-dependent parabolic f- alpha diagram due to a finite size effect. A possible systematic error in the conventional multifractal analysis is discussed.

Random magnetic fields acting on the boundary of a system are relevant if eta /sub ///(1. For the case of the two-dimensional Ising model they are marginal, giving rise to calculable logarithmic corrections to boundary quantities near the bulk critical point. These considerations may be important in the study of finite-size effects in the critical behaviour of adsorbed systems.

From the symplectic structure on the q-deformed manifold S_q² of S², the classical and quantum q-deformed algebras SU_{q,h(cross) to 0}(2) and SU_q,h(cross)(2) of SU(2) are discussed in detail and their Hopf structures are constructed. It is also shown that the symplectic geometry of S² describes the properties of monopoles for the SU(2) algebra. The q-(deformed) monopoles are studied from the symplectic geometry of S_q², and a relation between generic 2D manifold and algebraic structures is established.

The authors derive partial differential equations for characters of Kac-Moody algebras by using Ward identities for energy-momentum tensors and currents on a torus and the null vectors of current algebras. The solutions of the partial differential equations for affine algebras A⁽¹⁾_l, D⁽¹⁾_l, E⁽¹⁾_l (l=6, 7, 8) and B⁽¹⁾_l, at level one, are given.

A theorem concerning the explicit form of the eigenvalues of the class sums of the symmetric group (S_n) is derived and used to obtain the following results: (1) the centre of the S_n-algebra is generated by means of polynomials in the set of elements consisting of the generators of the centre of the S_n-k algebra augmented by the single cycle class sums ((2))_n, ((3))_n, . . ., ((k+1))_n; (2) the irreps of S_n with up to k rows are fully specified by the class sums ((2))_n, ((3))_n, . . ., ((k))_n. Furthermore, it is found that the k class sums ((2))_n, ((3))_n, . . ., ((k+1))_n suffice to specify the irreps of S_n for all n<or=n_max(k), where n_max(k)>>k.

It is shown that Darboux transformations can be used to find explicit solutions of deformed equations which have recently been found through the inverse method with a variable spectral parameter. The author constructs in particular the soliton solution of the deformed Maxwell-Bloch equations, and analyses its asymptotic behaviour.

The class of so-called Henon-Heiles systems is slightly broadened by allowing for the existence of non-standard Hamiltonians. The extra parameter in the equations of motion is shown to give rise to a generalization of the three known integrability cases. In addition, three degenerate cases are detected, characterized by a partial decoupling of the equations. For these cases, the author still obtains two independent first integrals, but their involutiveness can only be understood in terms of a nonstandard Poisson structure.

The Dirac analysis of constrained Hamiltonian mechanics is one of the conventional precursors to the quantization of classical systems. In this paper the analysis is reformulated in the language of exterior differential systems, starting from the Lagrangian, moving through the generation of primary and secondary constraints and leading to the construction of symmetry generators for gauge symmetries. This reformulation extends the procedure to non-coordinate systems. A computer algebra implementation of the procedure in REDUCE is also described.

It is shown that the Schrodinger equation with a linear confinement potential has an analytical solution. The existence of the analytical solution requires additional terms to the potential other than the linear one. The Coulomb-type potential is necessary to get non-zero energy eigenvalues. The author proposes an asymptotically linear potential, and shows that one can analytically solve the Schrodinger equation with this potential.

This is an attempt to clarify the structure of spontaneous symmetry breaking. It is shown that there are two types of situation. In one, called here SBS1, a symmetric ground state is degenerate with an asymmetric one; in the other, SBS2, the ground state belongs to a representation other than the identical one. Some cases which look like spontaneous symmetry breaking are in fact symmetry-breaking approximations.

The discrete spectrum of a q-analogue of the hydrogen atom is obtained from a deformation of the Pauli equations. As an alternative, the spectrum is derived from a deformation of the four-dimensional oscillator arising in the application of the Kustaanheimo-Stiefel transformation to the hydrogen atom. A model of the 2s-2p Dirac shift is proposed in the context of q-deformations.

The energy levels of the Schrodinger equation involving the potentials V(r)=r²+ lambda r²/(1+gr²) and V^-or+(r)=1/2(r²)-or+gr⁴/(1+g alpha r²) are calculated by using hypervirial and Pade approximant methods.

An algebraic method is used to solve the bound state problem for the most general Natanzon potential for which the Schrodinger equation can be reduced to the confluent hypergeometric form (hence, confluent potentials). The solution is obtained straightforwardly. A simple argument is given to sustain that the Natanzon potential is the most general confluent potential.

The authors study a spin glass hypercubic cell in D dimensions (i.e. hypercubic lattice with L=2 where L is the lattice size) for different values of D by means of Monte Carlo simulation. In the limit D to infinity this model converges to the SK model as well as for hypercubic lattices. They confirm the usual interpretation of the spin glass phase. Since this model is more similar to hypercubic lattices than to the infinite-range model these results suggest that broken replica symmetry is useful to study finite-dimensional hypercubic lattices above a certain critical dimension.

Derives a bosonic and fermionic field theory for randomly directed polymers. The bosonic field theory needs replicas and it is shown that the problem can be mapped in an O(n) symmetric Lagrangian with an attractive phi ⁴ term. The fermionic field theory introduces alternatively anticommutating fields and a supersymmetric Lagrangian is derived for the randomly directed polymer.

The authors show with the aid of a canonical transformation that separates charge and spin degrees of freedom that the strongly correlated one-dimensional Hubbard model with infinite (U= infinity ) on-site repulsion is related to a corresponding spinless free-fermion lattice model. Thermodynamic properties of the two models are shown to be closely related whereas the single-particle two-site function for the U= infinity model is shown to be related to a modified many site function for the spinless model which involves many-point correlations. The latter is shown to be expressible in terms of the inverse of a certain matrix which should be amenable to numerical and analytical analysis.

Characterization and measurement of fluctuations in quantum and classical populations are compared. A simple method for estimating the population fluctuation time is outlined and approximate quantum and classical expressions for the two-time correlation function are derived. The utility of the method is demonstrated through analysis of the Scully-Lamb laser model (1967), where it is shown to give correlation times in good agreement with numerical calculations for lasers operated below, at, and above threshold. Discrepancies in the predictions of a previous similar method are shown to originate from inconsistent separation of its quantum and classical versions.

The authors investigate by Monte Carlo simulations the thermodynamic behaviour of a linear heteropolymer in which the interaction between different monomers contains a quenched random component. They show the existence, along with the usual coil and globule phases, of a new folded phase, characterized by long relaxation times and by the existence of few stable states.

Two-dimensional site directed percolation is studied by the mean-field renormalization group approach to bulk and surface critical properties. Extrapolation techniques for the exponents and the percolation threshold are described. A very good estimate of the percolation threshold (p_c=0.7055+or-0.0001) and the first evaluation of the surface exponent y_hs=0.653+or-1 are obtained.

The critical curve with central charge c=-22/5 of the Ising quantum chain in an imaginary longitudinal field is calculated by finite-size extrapolation of the Yang-Lee square root branch point position. Various power laws are found. The field dependence of the off-criticality mass scale and the four lowest off-criticality scaling functions are calculated. The latter turn out to be universal along the critical line and agree very well with the results of the truncated Hilbert space method for the c=-22/5 continuum theory. The author obtains the universal bulk scaling coefficient in qualitative agreement with the thermodynamic Bethe ansatz result. The scattering phaseshift calculated by Luscher's method (1988) reproduces the Cardy-Mussardo minimal solution (1989) for the S-matrix. If a boundary defect is introduced, the only non-trivial primary field decouples, so that the spectrum consists only of the tower of the unit operator. In the antiferromagnetic regime with a non-staggered imaginary field the author finds boundary-dependent lines where the gap vanishes.

The authors examine the role played by dipolar interactions in the equilibrium critical behaviour of the dilute antiferromagnet in a uniform field (DAFF). The DAFF has been known to map onto a random field model, and we perform a similar mapping, only this time including dipolar interactions. They conclude that dipolar forces affect the DAFF in a way no different from the case for the corresponding pure system. Dilution does not generate any additional long-range forces or other effects arising from dipolar forces. In particular, in the final random field model which the DAFF maps onto, they find that the mean-field propagator is unaffected by the inclusion of dipolar interactions save for some constant terms. They restrict the work to the critical region close to the transition temperature and assume a domain-free state.

The exponent phi characterizing the divergence of diamagnetic susceptibility of a granular superconductor network in two dimensions is calculated using a low concentration series expansion, and found to be 1.21+or-0.03 as compared with phi =1.35 predicted from the scaling relation phi =2 nu -t and a recent simulation. This discrepancy is probably due to the shortness of the series.

A new approximate method for studying directed percolation in plane crystals is presented. By applying the method the percolation threshold and some statistical characteristics of the infinite cluster in a plane crystal with a quadratic lattice are found. The results obtained are verified by the Monte Carlo method.

The authors find that the dynamics of a dilute network of symmetric Hebb-rule synapses with parallel evolution, the model considered by Patrick and Zagrebnov (1990), are considerably more complicated than that paper predicted. The number of parameters is not constant but equal to the square of the number of time steps taken and they are generated by exact but increasingly complicated relations. A replica-symmetric calculation of long-term behaviour gives an estimate of alpha _c approximately=1.26. The calculations also describe parallel evolution of fully connected spin glasses.