Table of contents

In this letter we present a brief description of a ladder operator formalism applied to orthogonal polynomials with discontinuous weights. The two coefficient functions, A_n(z) and B_n(z), appearing in the ladder operators satisfy the two fundamental compatibility conditions previously derived for smooth weights. If the weight is a product of an absolutely continuous reference weight w₀ and a standard jump function, then A_n(z) and B_n(z) have apparent simple poles at these jumps. We exemplify the approach by taking w₀ to be the Hermite weight. For this simpler case we derive, without using the compatibility conditions, a pair of difference equations satisfied by the diagonal and off-diagonal recurrence coefficients for a fixed location of the jump. We also derive a pair of Toda evolution equations for the recurrence coefficients which, when combined with the difference equations, yields a particular Painlevé IV.

An anomalous diffusion equation is solved by the method of Lie symmetries. The nonlinear diffusion equation includes fractal dimensions and power-law dependence on the radial variable and the diffusion function. The Lie symmetries, appropriate boundary and initial conditions and a first integral replace an ansatz that was previously used to find a class of analytic solutions. The mean-squared displacement of the diffusion varies in time t by an exponent that is easily found by the Lie symmetries and boundary conditions.

We study the excited random walk, in which a walk that is at a site that contains cookies eats one cookie and then hops to the right with probability p and to the left with probability q = 1 − p. If the walk hops onto an empty site, there is no bias. For the 1-excited walk on the half-line (one cookie initially at each site), the probability of first returning to the starting point at time t scales as t^−(2−p). Although the average return time to the origin is infinite for all p, the walk eats, on average, only a finite number of cookies until this first return when p < 1/2. For the infinite line, the probability distribution for the 1-excited walk has an unusual anomaly at the origin. The positions of the leftmost and rightmost uneaten cookies can be accurately estimated by probabilistic arguments and their corresponding distributions have power-law singularities. The 2-excited walk on the infinite line exhibits peculiar features in the regime p > 3/4, where the walk is transient, including a mean displacement that grows as t^ν, with dependent on p, and a breakdown of scaling for the probability distribution of the walk.

We study the effect of magnetic impurity on the entanglement in the three-site Ising model with both periodic and open boundary conditions. We obtain the analytic expressions for entanglement and show that the ground state and excited states of the system possess remarkable entanglement properties. We find that the entanglement between two specific spins for a given state depends on the sites where the impurities are located. We also discuss a relation of our results with some aspects of quantum phase transitions in the transverse Ising model.

The Helfrich–Canham bending energy is identified with a nonlinear sigma model for a unit vector. The identification, however, is dependent on one additional constraint: that the unit vector be constrained to lie orthogonal to the surface. The presence of this constraint adds a source to the divergence of the stress tensor for this vector so that it is not conserved. The stress tensor which is conserved is identified and its conservation shown to reproduce the correct shape equation.

We consider higher-dimensional generalizations of the classical one-dimensional 2-automatic paperfolding and Rudin–Shapiro sequences on . This is done by considering the same automaton-structure as in the one-dimensional case, but using binary number systems in instead of in . The correlation function and the diffraction spectrum for the resulting m-dimensional paperfolding and Rudin–Shapiro point sets are calculated through the corresponding sequences with values ±1. They are shown to be quasi-independent of the dimension m and of the particular binary number system under consideration. It is shown that any paperfolding sequence thus obtained has a discrete spectrum. The Rudin–Shapiro sequences have an absolutely continuous Lebesgue spectral measure.

We show that, under twistor construction, the only finite-dimensional reduction of the dispersionless Dym hierarchy which can be characterized by a free energy is the two-reduction model with Lax operator of the form λ = up + u₀ + p⁻¹. The primary variables u and u₀ can be expressed in terms of second derivatives of the associated free energy and the hierarchy flows can be written as a set of dispersionless Hirota equations.

The aim of this paper is three-fold: first, to establish in a clear and rigorous way a formula proposed heuristically by Mehlig and Wilkinson for the metaplectic operators corresponding to a given symplectic transformation in classical phase space. Second, this formula is applied to the study of quantum recurrences, which has attracted a great deal of interest in recent years (see [35] for a complete account of the recent approaches). The return probability is given by the squared modulus of the overlap between a given initial wavepacket and the corresponding evolved one; quantum recurrences in time can be observed if this overlap is unity. We provide some conditions under which this is semiclassically achieved taking as the initial wavepacket a coherent state localized on a closed orbit of the corresponding classical motion. Third, we start a rigorous approach of the 'quantum fidelity' (or the Loschmidt echo): it is the squared modulus of the overlap of an evolved quantum state with the same state evolved by a slightly perturbed Hamiltonian. It has attracted a great deal of interest in the last decade, for the purpose of 'quantum chaos' problems, and in quantum computation analysis. However, the results are most of the time not entirely conclusive, and sometimes even contradictory. Thus it is useful to start a rigorous approach to this problem. The decrease in time of the quantum fidelity measures the sensitivity of quantum evolution with respect to small perturbations. Starting with suitable initial quantum states, we develop a semiclassical estimate of this quantum fidelity in the linear response framework (appropriate for the small perturbation regime), assuming some ergodicity conditions on the corresponding classical motion.

In this paper, we provide an explicit parametrization of arbitrary n-dimensional Hermitian operators on the Hilbert space , operators that may be considered either as Hamiltonians or density matrices for finite-level quantum systems, the description of which gives a complete solution to the over parametrization problem. It is shown that all the spectral multiplicities are encoded in a flag unitary matrix obtained as an ordered product of special unitary matrices, each one generated by a complex n − k-dimensional unit vector, k = 0, 1, ..., n − 2. As a byproduct, an alternative and simple parametrization of Stiefel and Grassmann manifolds is obtained.

We consider the C*-algebras and generated, respectively, by isometries s₁, s₂ satisfying the relation s*₁s₂ = qs₂s*₁ with |q| < 1 (the deformed Cuntz relation), and by isometries s₁, s₂ satisfying the relation s₂s₁ = qs₁s₂ with |q| = 1. We show that is isomorphic to the Cuntz–Toeplitz C*-algebra for any |q| < 1. We further prove that if and only if either q₁ = q₂ or . In the second part of our paper, we discuss the complexity of the representation theory of . We show that is *-wild for any q in the circle |q| = 1, and hence that is not nuclear for any q in the circle.

The well-known Lorenz system can be written as and . Here, we study the first integrals of the Lorenz system that can be described by formal power series. In particular, if s ≠ 0 and, either b is not a negative rational number, or b is a negative rational number and k₁b + k₂(1 + s) ≠ 0, for all k₁ and k₂ non-negative integers with k₁ + k₂ > 0, then the Lorenz system has no analytic first integrals in a neighbourhood of the origin.

A nilpotent Lie algebra with an (n − 1)-dimensional Abelian ideal is studied. All indecomposable solvable Lie algebras with as their nilradical are obtained. Their dimension is at most n + 2. The generalized Casimir invariants of and of its solvable extensions are calculated. For n = 4 these algebras figure in the Petrov classification of Einstein spaces. For larger values of n they can be used in a more general classification of Riemannian manifolds.

Using a combination of exact results from diffraction theory and the method of images, we calculate to leading order in the high-frequency limit, the modes excited within a waveguide comprising a pair of semi-infinite parallel half-planes when a wave strikes its aperture. We consider several types of incident field, and the calculation that we present does not rely on any approximations (such as stationary phase or steepest descent integral evaluations) and is somewhat less involved than methods in the existing literature that do so. Furthermore, it permits the consideration of situations in which 'Fresnel' regions propagate within the guide, and of more complicated geometries, whilst retaining its straightforward nature.

We give details of calculations analysing the proposed mirror superposition experiment of Marshall, Simon, Penrose and Bouwmeester within different stochastic models for state vector collapse. We give two methods for exactly calculating the fringe visibility in these models, one proceeding directly from the equation of motion for the expectation of the density matrix, and the other proceeding from solving a linear stochastic unravelling of this equation. We also give details of the calculation that identifies the stochasticity parameter implied by the small-displacement Taylor expansion of the CSL model density matrix equation. The implications of the two results are briefly discussed. Two pedagogical appendices review mathematical apparatus needed for the calculations.

Careful monitoring of harmonically bound (or as a limiting case, free) masses is the basis of current and future gravitational wave detectors, and of nanomechanical devices designed to access the quantum regime. We analyse the effects of stochastic localization models for state vector reduction, and of related models for environmental decoherence, on such systems, focusing our analysis on the non-dissipative forced harmonic oscillator and its free mass limit. We derive an explicit formula for the time evolution of the expectation of a general operator in the presence of stochastic reduction or environmentally induced decoherence, for both the non-dissipative harmonic oscillator and the free mass. In the case of the oscillator, we also give a formula for the time evolution of the matrix element of the stochastic expectation density matrix between general coherent states. We show that the stochastic expectation of the variance of a Hermitian operator in any unraveling of the stochastic process is bounded by the variance computed from the stochastic expectation of the density matrix, and we develop a formal perturbation theory for calculating expectation values of operators within any unraveling. Applying our results to current gravitational wave interferometer detectors and nanomechanical systems, we conclude that the deviations from quantum mechanics predicted by the continuous spontaneous localization (CSL) model of state vector reduction are at least five orders of magnitude below the relevant standard quantum limits for these experiments. The proposed LISA gravitational wave detector will be two orders of magnitude away from the capability of observing an effect.

A complete set of d + 1 mutually unbiased bases exists in a Hilbert space of dimension d, whenever d is a power of a prime. We discuss a simple construction of d + 1 disjoint classes (each one having d − 1 commuting operators) such that the corresponding eigenstates form sets of unbiased bases. Such a construction works properly for prime dimension. We investigate an alternative construction in which the real numbers that label the classes are replaced by a finite field having d elements. One of these classes is diagonal, and can be mapped to cyclic operators by means of the finite Fourier transform, which allows one to understand complementarity in a similar way as for the position–momentum pair in standard quantum mechanics. The relevant examples of two and three qubits and two qutrits are discussed in detail.

We study the entanglement dynamics of the three-qubit system in a symmetry-broken environment consisting of a fermionic bath. Decoherence induced by the bath is analysed. We find that the entanglement of states will decrease or remain unchanged under the system–bath interaction. The class of decoherence-free states of the three-qubit system of our model has been found out.

Using an algebraic orbifold method, we present non-commutative aspects of G₂ structure of seven-dimensional real manifolds. We first develop and solve the non-commutativity parameter constraint equations defining G₂ manifold algebras. We show that there are eight possible solutions for this extended structure, one of which corresponds to the commutative case. Then, we obtain a matrix representation solving such algebras using combinatorial arguments. An application to matrix model of M-theory is discussed.

We investigate a model of interacting charged particles in two space dimensions, with manifest invariance under duality and periodicity under flux attachment. This model, introduced by Fradkin and Kivelson (1996 Nucl. Phys. B 474 543), shares many qualitative features of real quantum Hall systems. We extend this model to the case of finite filling fraction, i.e., to physical systems without particle–hole symmetry and without time-reversal invariance. We derive the transformation laws for the the average currents and prove that they have an SL (2, Z) symmetry. We can then calculate the filling factors at the modular fixed points and further explore the topological order of the model by constructing the hierarchy of states.

The classical nonrelativistic motion of an electric charge in the presence of a magnetic monopole is reviewed. Using general properties of these trajectories it is shown that the path integral form of the WKB approximation for the charged particle propagator develops an anomalous form for the transition probability density unless the Dirac quantization condition is imposed. This result is shown to be independent of the specific form of the vector potential used to represent the magnetic monopole. The general result is then verified for the specific case that the standard Dirac string solution is used in the path integral. This result is interpreted as an indirect demonstration of the Dirac condition. Other properties of the solutions and the associated action are briefly discussed.

The nonlinear Kelvin–Helmholtz stability of the interface between the vapour and liquid phases of a fluid is studied when the phases are confined between two parallel lines, and when there is mass and heat transfer across the interface. The method of multiple time expansion is used for the investigation. The evolution of amplitude is shown to be governed by a nonlinear first-order differential equation. The stability criterion is discussed, and the region of stability is displayed graphically.

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