Table of contents

We formulate a systematic algorithm for constructing a whole class of Hermitian position-dependent-mass Hamiltonians which, to lowest order of perturbation theory, allow a description in terms of -symmetric Hamiltonians. The method is applied to the Hermitian analogue of the -symmetric cubic anharmonic oscillator. A new example is provided by a Hamiltonian (approximately) equivalent to a -symmetric extension of the one-parameter trigonometric Pöschl–Teller potential.

In this paper we consider flow equations where we allow a normal ordering which is adjusted to the one-particle energy of the Hamiltonian. We show that this flow nearly always converges to the stable phase. Starting out from the symmetric Hamiltonian and symmetry-broken normal ordering nearly always yields symmetry breaking below the critical temperature.

We consider the ground state of a system of chargeless fermions, such as neutrinos, of mass m and magnetic moment μ interacting through long-range magnetic dipole interaction, within the framework of a Hartree–Fock variational approach. At high densities the uniform paramagnetic state becomes unstable towards a ferromagnetic state with quadrupolar deformation of the Fermi surface. The exchange energy which is attractive dominates the repulsive kinetic energy. If we let the density be a variable, then above a certain density the system will collapse to an infinite density state unless another short-range interaction stops the collapse. In the case of large deformations, the possibility of a purely dipolar deformation exists.

Density waves are investigated in the full velocity difference model (FVDM) analytically and numerically. By the use of nonlinear analysis, the Burgers, Korteweg–de Vries (KdV) and Modified KdV equations are derived for the triangular shock wave, the soliton wave and the kink–antikink wave, respectively, appearing in the stable region out of the coexisting curve, near the spinodal line, and in the unstable region within the spinodal line. It is shown, numerically, that the triangular shock wave and the soliton wave are determined by the initial perturbation configuration and different initial perturbations will produce different waveforms in the stable region or near the spinodal line.

In this paper we investigate a finite temperature generalization of survey propagation, by applying it to the problem of finite temperature decoding of a biased finite connectivity Sourlas code for temperatures lower than the Nishimori temperature. We observe that the result is a shift of the location of the dynamical critical channel noise to larger values than the corresponding dynamical transition for belief propagation, as suggested recently by Migliorini and Saad for LDPC codes. We show how the finite temperature 1RSB SP gives accurate results in the regime where competing approaches fail to converge or fail to recover the retrieval state.

We consider a deterministic realization of Parrondo games, and use periodic orbit theory to analyse their asymptotic behaviour.

We investigate the statistics of a vector Manakov soliton in the presence of additive Gaussian white noise. The adiabatic perturbation theory for a Manakov soliton yields a stochastic Langevin system which we analyse via the corresponding Fokker–Planck equation for the probability density function (PDF) for the soliton parameters. We obtain marginal PDFs for the soliton frequency and amplitude as well as soliton amplitude and polarization angle. We also derive formulae for the variances of all soliton parameters and analyse their dependence on the initial values of polarization angle and phase.

We examine the transport behaviour of non-interacting particles in a simple channel billiard, at equilibrium and in the presence of an external field. We observe a range of sub-diffusive, diffusive and super-diffusive transport behaviours. We find nonequilibrium transport that is inconsistent with the equilibrium behaviour, indicating that a linear regime does not exist for this system. However, we define a 'weak' linear regime that may lead to consistency between the equilibrium and nonequilibrium results. Despite the non-chaotic nature of the dynamics, we observe greater unpredictability (complexity) in the transport properties than is observed for chaotic systems. This observation seems at odds with existing complexity measures, motivating some new measures that attribute importance to the collective behaviour, rather than to the behaviour of individual system components. Such measures provide a new key for characterizing complexity in real-world phenomena.

The indecomposable solvable Lie algebras with graded nilradical of maximal nilindex and a Heisenberg subalgebra of codimension one are analysed, and their generalized Casimir invariants are calculated. It is shown that rank one solvable algebras have a contact form, which implies the existence of an associated dynamical system. Moreover, due to the structure of the quadratic Casimir operator of the nilradical, these algebras contain a maximal non-abelian quasi-classical Lie algebra of dimension 2n − 1, indicating that gauge theories (with ghosts) are possible on these subalgebras.

The path integral treatment of the step potential based on exact summation of the Feynman perturbation series is presented in its original form. With a two-sided Laplace transform on the end point, we have established a recurrence formula between three successive terms of the perturbation series which leads us to sum up exactly all the terms and deduce the Green function of the problem. The Wiener–Hopf method and Hilbert problems are successfully used for solving our problem with the boundary conditions.

The set of points of a one-dimensional cut-and-project quasicrystal or model set, while not additive, is shown to be multiplicative for appropriate choices of acceptance windows. This leads to the definition of an associative additive graded composition law and permits the introduction of Lie algebras over such aperiodic point sets. These infinite-dimensional Lie algebras are shown to be representatives of a new type of semi-direct product induced Lie algebras.

We consider a spin 1/2 charged particle on the Lobachevsky plane subjected to a magnetic field corresponding to a discrete system of Aharonov–Bohm solenoids. Let H⁺ and H⁻ be the two components of the Pauli operator for spin-up and -down, respectively. We show that neither H⁺ nor H⁻ has a zero mode if the number of solenoids is finite. On the other hand, a construction is described of an infinite periodic system of solenoids for which either H⁺ or H⁻ has zero modes depending on the value of the flux carried by the solenoids.

Satake diagrams of affine Kac–Moody algebras (untwisted and twisted) are obtained from their Dynkin diagrams. These diagrams give a classification of restricted root systems associated with these algebras. In the case of simple Lie algebras, these root systems and Satake diagrams correspond to symmetric spaces which have recently found many physical applications in quantum integrable systems, quantum transport problems, random matrix theories etc. We hope these types of root systems may have similar applications in theoretical physics in future and may correspond to symmetric spaces analogue of affine Kac–Moody algebras if they exist.

We give a detailed analysis of the anti-self-adjoint operator contribution to the fluctuation terms in the trace dynamics Ward identity. This clarifies the origin of the apparent inconsistency between two forms of this identity discussed in chapter 6 of our recent book on emergent quantum theory.

Drawing inspiration from Dirac's work on functions of non-commuting observables, we develop an approach to phase-space descriptions of operators and the Wigner–Weyl correspondence in quantum mechanics, complementary to standard formulations. This involves a two-step process: introducing phase-space descriptions based on placing position dependences to the left of momentum dependences (or the other way around); then carrying out a natural transformation to eliminate a kernel which appears in the expression for the trace of the product of two operators. The method works uniformly for both continuous Cartesian degrees of freedom and for systems with finite-dimensional state spaces. It is interesting that the kernel encountered is naturally expressible in terms of geometric phases, and its removal involves extracting its square root in a suitable manner.

Very recently, an exponential probability distribution with parameter has been used to calculate the decoherence factors for quantum states. We derive all moments of this distribution systematically in three different ways, presenting the results in terms of binomial coefficients or Pochhammer symbols, and Stirling numbers of the first and second kinds. We show how generalized harmonic numbers or polygamma functions provide another representation for the moments. Extensions of the approach are briefly mentioned.

m-qubit states are embedded in Clifford algebras. Their probability spectrum then depends on O(2m)- or O(2m + 1)-invariants, respectively. Parameter domains for O(2m(+1))-vector and -tensor configurations, generalizing the notion of a Bloch sphere, are derived.

With the help of the one-dimensional quintic complex Ginzburg–Landau equation (CGLE) as perturbations of the nonlinear Schrödinger equation (NLSE), we derive the equations of motion of pulse parameters called collective variables (CVs), of a pulse propagating in dispersion-managed (DM) fibre optic links. The equations obtained are investigated numerically in order to view the evolution of pulse parameters along the propagation distance, and also to analyse effects of initial amplitude and width on the propagating pulse. Nonlinear gain is shown to be beneficial in stabilizing DM solitons. A fully numerical simulation of the one-dimensional quintic CGLE as perturbations of NLSE finally tests the results of the CV theory. A good agreement is observed between both methods.

Quantum gates are represented by unitary operators on a Hilbert space. We consider the unitary operators on and describe the component operators on each Hilbert space in the product space. From these criteria we derive a decomposition for a specific class of unitary operators such that each operator in the product has Schmidt rank bounded by 2. This decomposition directly yields the XOR (controlled NOT) implementation of the SWAP operation.

Theories of Torres-Vega and Fredrick (1993 J. Chem. Phys.98 3103), Harriman (1994 J. Chem. Phys.100 3651) and of Ban (1998 J. Math. Phys.39 1744), in which phase space points (p, q) are used as configurational variables to formulate quantum mechanics are considered from the standpoint of a class of quantization schemes associating phase space functions with operators. The connection between these schemes and the theories given in Torres-Vega and Fredrick (1993 J. Chem. Phys.98 3103), Harriman (1994 J. Chem. Phys.100 3651), Dirac (1930 The Principles of Quantum Mechanics (Oxford: Oxford University Press)), Møller, Jørgensen and Torres-Vega (1997 J. Chem. Phys.106 7228), Klauder and Skagerstam (1985 Coherent States: Applications in Physics and Mathematical Physics (Singapore: World Scientific)), Li, Wei and Lü (2004 Phys. Rev. A 70 022105), Ban (1998 J. Math. Phys.39 1744) is made by means of augmented wavefunctions ψ^(λ)_σ(p, q; t), where λ = 0 corresponds to the ordering of Wigner and Weyl. For that case we use these functions to define a family of positive operator-valued measures for the phase angle of an harmonic oscillator.

Homologous operational localization processes are effectuated in terms of generalized topological covering systems on structures of physical events. We study localization systems of quantum events' structures by means of Gtothendieck topologies on the base category of Boolean events' algebras. We show that a quantum events algebra is represented by means of a Grothendieck sheaf-theoretic fibred structure, with respect to the global partial order of quantum events' fibres over the base category of local Boolean frames.

Quantized Skyrmions with baryon numbers B = 1, 2 and 4 are considered and angularly localized wavefunctions for them are found. By combining a few low angular momentum states, one can construct a quantum state whose spatial density is close to that of the classical Skyrmion and has the same symmetries. For the B = 1 case, we find the best localized wavefunction among linear combinations of and angular momentum states. For B = 2, we find that the j = 1 ground state has toroidal symmetry and a somewhat reduced localization compared to the classical solution. For B = 4, where the classical Skyrmion has cubic symmetry, we construct cubically symmetric quantum states by combining the j = 0 ground state with the lowest rotationally excited j = 4 state. We use the rational map approximation to compare the classical and quantum baryon densities in the B = 2 and B = 4 cases.

The spatial evolution of vector electromagnetic fields in bianisotropic cylindrically symmetric structures is studied. We introduce evolution operators (characteristic matrices), impedance tensors, reflection and transmission operators of cylindrical beams which are convenient for describing waves in layered media. We predict the existence of fractional Bessel beams with integer topological charge and cylindrically symmetric beams which can be presented as the product of the Bessel function and the exponent. We investigate energy and polarization characteristics of such beams.

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