Table of contents

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Neil Scriven Publisher Journal of Physics A: Mathematical and Generalneil.scriven@iop.org

We study the Blume-Emery-Griffiths spin-glass model in the presence of an attractive coupling between real replicas, and evaluate the effective potential as a function of the density overlap. We find that there is a region, above the first-order transition of the model, where metastable states with a large density overlap exist. The line where these metastable states appear should correspond to a purely dynamical transition, with a breaking of ergodicity. Differently from what happens in p-spin glasses, in this model the dynamical transition would not be the precursor of a one-step replica symmetry breaking transition, but (probably) of a full replica symmetry breaking transition.

We propose an unprecedented bounding theory which generates converging bounds to the Regge poles of rational fraction scattering potentials. This is made possible by the recent work of Handy ( 2001 J. Phys. A: Math. Gen.34 L271) and Handy and Wang (2001 J. Phys. A: Math. Gen.34 8297) which transforms the Schrödinger equation into an equivalent fourth-order, lineardifferential equation for the probability density. This new representation is better suited for numerical considerations, since the rapid oscillations of the Regge-pole wavefunction are factored out. More importantly, the moments of the probability density can be constrained (and thereby the underlying complex angular momentum parameter of the effective potential function) through appropriate moment problem theorems, as incorporated within the eigenvalue moment method of Handy and Bessis (1985 Phys. Rev. Lett.55 931) and Handy et al (1988 Phys. Rev. Lett.60 253).

We study the dynamics of the `batch' minority game with market-impact correction using generating functional techniques to carry out the quenched disorder average. We find that the assumption of weak long-term memory, which one usually makes in order to calculate ergodic stationary states, breaks down when the persistent autocorrelation becomes larger than c_c≃0.772. We show that this condition, remarkably, coincides with the AT line found in an earlier static calculation. This result suggests a new scenario for ergodicity breaking in disordered systems.

We investigate cascades of isochronous pitchfork bifurcations of straight-line librating orbits in some two-dimensional Hamiltonian systems with mixed phase space. We show that the new bifurcated orbits, which are responsible for the onset of chaos, are given analytically by the periodic solutions of the Lamé equation as classified in 1940 by Ince. In Hamiltonians with C_2v symmetry, they occur alternatingly as Lamé functions of period 2K and 4K, respectively, where 4K is the period of the Jacobi elliptic function appearing in the Lamé equation. We also show that the two pairs of orbits created at period-doubling bifurcations of island-chain type are given by two different linear combinations of algebraic Lamé functions with period 8K.

We study corrections to scaling in the O(3)- and O(4)-symmetric ϕ⁴ model on the three-dimensional simple cubic lattice with nearest-neighbour interactions. For this purpose, we use Monte Carlo simulations in connection with a finite-size scaling method. We find that a finite value of the coupling λ* exists for both values of N, where leading corrections to scaling vanish. As a first application, we compute the critical exponents ν = 0.710(2) and η = 0.0380(10) for N = 3 and ν = 0.749(2) and η = 0.0365(10) for N = 4.

One of the central problems in the representation theory of compact groups concerns multiplicity, wherein an irreducible representation occurs more than once in the decomposition of the n-fold tensor product of irreducible representations. The problem is that there are no operators arising from the group itself whose eigenvalues can be used to label the equivalent representations occurring in the decomposition.

In this paper we use invariant theory along with so-called generalized Casimir operators to show how to resolve the multiplicity problem for the U(N) groups. The starting point is to augment the n-fold tensor product space with the contragredient representation of interest and construct a subspace of U(N) invariants. The setting for this construction is a polynomial space embedded in a Fock space of complex variables which carries all the irreducible representations of U(N) (or GL_N ). The dimension of the invariant subspace is equal to the multiplicity occurring in the tensor product decomposition.

Generalized Casimir operators are operators from the universal enveloping algebra of outer product U(N) groups that commute with the diagonal U(N) action and whose eigenvalues can be used to label the multiplicity. Using the notion of dual representations we show how to rewrite the generalized Casimir operators and prove that they act invariantly on the invariant subspace. A complete set of commuting generalized Casimir operators can therefore be used to construct eigenvectors that form an orthonormal basis in the invariant subspace. Different sets of generalized commuting Casimir operators generate different orthonormal bases in the invariant subspace; the overlaps between the eigenvectors of different commuting sets of generalized Casimir operators are called invariant coefficients. We show that Racah coefficients are special cases of invariant coefficients in which the generalized Casimir operators have been chosen with respect to a definite coupling scheme in the tensor product.

The paper concludes with an example of the threefold tensor product of the eight-dimensional irreducible representation of U(3) in which the multiplicity of the chosen irreducible representation is 6. Eigenvectors in the six-dimensional invariant subspace are computed for different sets of generalized Casimir operators and invariant coefficients, including Racah coefficients.

We present an integrable Hamiltonian which describes the sinh-Gordon model on the half line coupled to a non-linear oscillator at the boundary. We explain how we apply Sklyanin's formalism to a dynamical reflection matrix to obtain this model. This method can be applied to couple other integrable field theories to dynamical systems at the boundary. We also show how to find the dynamical solution of the quantum reflection equation corresponding to our particular example.

The eigenvalues of the potentials

V₁(r) = A₁/r + A₂/r² + A₃/r³ + A₄/r⁴

and

V₂(r) = B₁r² + B₂/r² + B₃/r⁴ + B₄/r⁶

and of the special cases of these potentials such as Kratzer and Goldman-Krivchenkov potentials, are obtained in N-dimensional space. The explicit dependence of these potentials in higher-dimensional space is discussed, which has not been previously covered.

This article presents a new method to calculate eigenvalues of right triangle billiards. Its efficiency is comparable to the boundary integral method and more recently developed variants. Its simplicity and explicitness, however, allow new insight into the statistical properties of the spectra. We analyse numerically the correlations in level sequences at high level numbers (>10⁵) for several examples of right triangle billiards. We find that the strength of the correlations is closely related to the genus of the invariant surface of the classical billiard flow. Surprisingly, the genus plays an important role at the quantum level also. Based on this observation, a mechanism is discussed which may explain the particular quantum–classical correspondence in right triangle billiards. Though this class of systems is rather small, it contains examples for integrable, pseudo-integrable, and non-integrable (ergodic, weakly mixing) dynamics, so that the results might be relevant in a more general context.

We show that a recently developed method for generating bounds for the discrete energy states of the non-Hermitian -ix³ potential (Handy C R 2001 J. Phys. A: Math. Gen.34 L271) is applicable to complex rotated versions of the Hamiltonian. This has important implications for extension of the method in the analysis of resonant states, Regge poles, and general bound states in the complex plane (Bender and Boettcher 1998 Phys. Rev. Lett.80 5243).

A wide range of topics in the area of hypervirial perturbation theory (HVPT) is discussed. It is shown that with the use of a few simple procedures HVPT is capable of high accuracy for many problems; results from many previous works in the literature are found to be improvable by the careful use of HVPT with appropriate choice of the unperturbed potential and of the origin at which the energy expansion is carried out. Two multi-well problems from the literature are analysed in detail to show the value of a combination of HVPT and finite-difference methods.

A general framework is presented which unifies the treatment of wavelet-like, quasidistribution and tomographic transforms. Explicit formulae relating the three types of transforms are obtained. The case of transforms associated with the symplectic and affine groups is treated in some detail. Special emphasis is given to the properties of the scale-time and scale-frequency tomograms. Tomograms are interpreted as a tool to sample the signal space by a family of curves or as the matrix element of a projector.

Young tableaux are used to label the basis vectors of the standard or Young-Yamanouchi basis of the symmetric group. Despite being used for this purpose for some time, a physical interpretation of what they mean has not been given. Weyl tableaux however, which label the basis vectors of the standard or Gelfand basis of the unitary group, do have a physical interpretation. Weyl tableaux correspond to antisymmetrized states with definite total spin, definite spin projection and definite total angular momentum projection. We discuss how a previously well established link between Young and Weyl tableaux may imply Young tableaux are similarly associated with such states. If such an association could be made, calculations using symmetric group bases could be reduced to simple angular momentum manipulations. In particular this would greatly increase the efficiency of calculating the coefficients of fractional parentage of S_n, which can in turn be used to calculate unitary recoupling coefficients. We present a methodology for determining whether such an association exists.

We describe an algebraic framework for studying the symmetry properties of integrable quantum systems on the half line. The approach is based on the introduction of boundary operators. It turns out that these operators both encode the boundary conditions and generate integrals of motion. We use this direct relationship between boundary conditions and symmetry content to establish the spontaneous breakdown of some internal symmetries, due to the boundary.

With a view towards future applications in nuclear physics, the fermion realization of the compact symplectic sp(4) algebra and its q-deformed versions are investigated. Three important reduction chains of the sp(4) algebra are explored in both the classical and deformed cases. The deformed realizations are based on distinct deformations of the fermion creation and annihilation operators. For the primary reduction, the su(2) substructure can be interpreted as either the spin, isospin or angular momentum algebra, whereas for the other two reductions su(2) can be associated with pairing between fermions of the same type or pairing between two distinct fermion types. Each reduction provides for a complete classification of the basis states. The deformed induced u(2) representations are reducible in the action spaces of sp(4) and are decomposed into irreducible representations.

We constructed the one-particle Green functions for two systems exhibiting giant magnetoresistance. The first one is a multilayer with arbitrary magnetization directions of the ferromagnetic layers, exchange splitting of the conducting electron band and intrinsic potential. The second one is a segmented nanowire with spin-dependent electron scattering at the lateral interfaces.