Table of contents

The generalized Swanson Hamiltonian with can be transformed into an equivalent Hermitian Hamiltonian with the help of a similarity transformation. It is shown that the equivalent Hermitian Hamiltonian can be further transformed into the harmonic oscillator Hamiltonian so long as constant. However, the main objective of this communication is to show that though the commutator of and is constant, the generalized Swanson Hamiltonian is not necessarily isospectral to the harmonic oscillator. The reason for this anomaly is discussed in the framework of position-dependent mass models by choosing A(x) as the inverse square root of the mass function.

We calculate the diffusion coefficients of persistent random walks on cubic and hypercubic lattices, where the direction of a walker at a given step depends on the memory of one or two previous steps. These results are then applied to study a billiard model, namely a three-dimensional periodic Lorentz gas. The geometry of the model is studied in order to find the regimes in which it exhibits normal diffusion. In this regime, we calculate numerically the transition probabilities between cells to compare the persistent random-walk approximation with simulation results for the diffusion coefficient.

Recently, Delfino and Viti have examined the factorization of the three-point density correlation function P₃ at the percolation point in terms of the two-point density correlation functions P₂. According to conformal invariance, this factorization is exact on the infinite plane, such that the ratio R(z₁, z₂, z₃) = P₃(z₁, z₂, z₃)/[P₂(z₁, z₂)P₂(z₁, z₃)P₂(z₂, z₃)]^1/2 is not only universal but also a constant, independent of z_i and in fact an operator product expansion coefficient. Delfino and Viti analytically calculated its value (1.022 013...) for percolation, in agreement with the numerical value 1.022 found previously in a study of R on the conformally equivalent cylinder. In this paper we confirm the factorization on the plane numerically using periodic lattices (tori) of very large size, which locally approximate a plane. We also investigate the general behavior of R on the torus, and find a minimum value of R ≈ 1.0132 when the three points are maximally separated. In addition, we present a simplified expression for R on the plane as a function of the SLE parameter κ.

The well-known scaling of the Edwards–Wilkinson equation is essentially determined by dimensional analysis. In a range of experimental setups, be it due to the presence of an electrical or a gravitational field, or the indirect effect of other terms or an expansion, an additional drift term has to be considered. Once the drift term is added, more sophisticated reasoning is required to determine the scaling, which initially suggests that the drift term dominates the diffusion. However, the diffusion term is dangerously irrelevant and the resulting scaling in fact non-trivial. In order to assess the universality of the Edwards–Wilkinson equation with drift and to describe a physically more relevant situation, we compare the scaling obtained with Neumann boundary conditions to the published case with Dirichlet boundary conditions.

We consider a model consisting of a self-avoiding polygon occupying a variable density of the sites of a square lattice. A fixed energy is associated with each 90° bend of the polygon. We use a grand canonical ensemble, introducing parameters μ and β to control average density and average (total) energy of the polygon, and show by Monte Carlo simulation that the model has a first order, nematic phase transition across a curve in the β–μ plane.

We investigate the phase diagram of a three-component system of particles on a one-dimensional filled lattice, or equivalently of a one-dimensional three-state Potts model, with reflection asymmetric mean-field interactions. The three types of particles are designated as A, B and C. The system is described by a grand canonical ensemble with temperature T and chemical potentials Tλ_A, Tλ_B and Tλ_C. We find that for λ_A = λ_B = λ_C the system undergoes a phase transition from a uniform density to a continuum of phases at a critical temperature . For other values of the chemical potentials the system has a unique equilibrium state. As is the case for the canonical ensemble for this ABC model, the grand canonical ensemble is the stationary measure satisfying detailed balance for a natural dynamics. We note that , where T_c is the critical temperature for a similar transition in the canonical ensemble at fixed equal densities r_A = r_B = r_C = 1/3.

The generalized multispin Jahn–Teller model on a finite lattice or formally equivalent Dicke model extended to two long-wavelength coherent bosons of different frequencies is shown to exhibit a crossover between the polaron-modified 'quasi-normal' and the squeezed 'radiation' domains. We investigate the effects of two kinds of interfering fluctuations on the phase crossover and on statistical characteristics of boson complex spectra. (i) Fluctuations in the electron subsystem—finite-size quantum fluctuations—are responsible for the dephasing of the coherence in the radiation domain and for the moderate occupation of the excited states in the normal domain. In the quasiclassical limit, radiation phase implies the existence of a coherent acoustic super-radiant phase. (ii) Level-spacing fluctuations in the excited boson level subsystem with strong level repulsions. Related probability distributions are shown to be non-universally spread between the limiting universal Wigner–Dyson and Poisson distributions. We proved that the difference in boson frequencies is responsible for reaching the most stochastic limit of the Wigner–Dyson distribution. Instanton lattice as a sequence of tunneling events in the most chaotic radiation domain exhibits maximal number of level-avoidings (repulsions). The non-universality of the distributions is caused by boson correlations which compete with the level repulsions.

We study probability distributions of eigenvalues of Hermitian and non-Hermitian Euclidean random matrices that are typically encountered in the problems of wave propagation in random media.

We first derive an integrable deformed hierarchy of a short pulse equation and its Lax representation. Then we concentrate on the solution of the integrable deformed short pulse equation (IDSPE). By proposing a generalized reciprocal transformation, we find a new integrable deformed sine-Gordon equation (IDSGE) and its Lax representation. The multisoliton solutions, negaton solutions and positon solutions for the IDSGE and the N-loop soliton solutions, N-negaton and N-positon solutions for the IDSPE are presented. In the reduced case the new N-positon solutions and N-negaton solutions for the short pulse equation are obtained.

Our concern is the study of the limit cycles emerging at a Hopf bifurcation. To do so, we consider an averaging method which allows us to find asymptotic expansions of the radius and frequency of the cycles. We consider a discriminant function where each root corresponds to an emerging cycle. A way of classifying the Hopf bifurcation on the basis of the number of emerging limit cycles arises from the present investigation: namely non-degenerate, first-type degenerate and second-type degenerate. We give genericity conditions for non-degenerate bifurcations up to 6-jet-equivalence, 'typical' forms for degenerate bifurcations, and also a sufficient condition for second-type degenerate bifurcation up to 6-jet-equivalence.

Unique transformation properties under the hyperspherical inversion of a partial differential equation describing a stationary scalar wave in an N-dimensional (N ⩾ 2) Maxwell fish-eye medium are exploited to construct a closed form of Green's function for that equation. For those wave numbers for which Green's function fails to exist, the generalized Green's function is derived. Prospective physical applications are mentioned.

In the framework of the Lagrangian formalism the partial differential equation under study does not define univocally the Lagrangian density. In this paper we obtain a necessary and sufficient consistency condition over the ansatz that assures the invariance of the collective coordinates (CCs) equations under the change of equivalent Lagrangian densities. When this condition is not fulfilled we show explicitly that different equations of CCs may emerge from equivalent Lagrangian densities and a good agreement between the CCs and the partial differential equations is not expected.

We explore quantum uncertainty relations involving the Fisher information functionals I_x and I_p evaluated, respectively, on a wavefunction Ψ(x) defined on a D-dimensional configuration space and the concomitant wavefunction on the conjugate momentum space. We prove that the associated Fisher functionals obey the uncertainty relation I_xI_p ⩾ 4D² when either Ψ(x) or is real. On the other hand, there is no lower bound to the above product for arbitrary complex wavefunctions. We give explicit examples of complex wavefunctions not obeying the above bound. In particular, we provide a parametrized wavefunction for which the product I_xI_p can be made arbitrarily small.

Using a quantum version of the Arnold transformation of classical mechanics, all quantum dynamical systems whose classical equations of motion are non-homogeneous linear second-order ordinary differential equations, including systems with friction linear in velocity, can be related to the quantum free-particle dynamical system. This transformation provides a basic (Heisenberg–Weyl) algebra of quantum operators, along with well-defined Hermitian operators which can be chosen as evolution-like observables and complete the entire Schrödinger algebra. It also proves to be very helpful in performing certain computations quickly, to obtain, for example, wavefunctions and closed analytic expressions for time-evolution operators.

The no-signaling polytope associated with a Bell scenario with three parties, two inputs, and two outputs, is found to have 53 856 extremal points, belonging to 46 inequivalent classes. We provide a classification of these points according to various definitions of multipartite nonlocality and briefly discuss other issues such as the interconversion between extremal points seen as a resource and the relation of the extremal points to Bell-type inequalities.

We compare entanglement with quantum nonlocality employing a geometric structure of the state space of bipartite qudits. The central object is a regular simplex spanned by generalized Bell states. The Collins–Gisin–Linden–Massar–Popescu–Bell inequality is used to reveal states of this set that cannot be described by local-realistic theories. Optimal measurement settings necessary to ascertain nonlocality are determined by means of a recently proposed parameterization of the unitary group combined with robust numerical methods. The main results of this paper are descriptive geometric illustrations of the state space that emphasize the difference between entanglement and quantum nonlocality. Namely, it is found that the shape of the boundaries of separability and Bell inequality violation are essentially different. Moreover, it is also shown that for mixtures of states sharing the same amount of entanglement, Bell inequality violations and entanglement measures are non-monotonically related.

We discuss a scheme for simulating the real-time quantum quench dynamics of interacting quantum spin systems within the positive-P formalism. As model systems we study the transverse field Ising model as well as the Heisenberg model undergoing a quench away from the classical ferromagnetic ordered state and the antiferromagnetic Néel state, depending on the sign of the Heisenberg exchange interaction. The connection to the positive-P formalism as it is used in quantum optics is established by mapping the spin operators on to Schwinger bosons. In doing so, the dynamics of the interacting quantum spin system is mapped onto a set of Ito stochastic differential equations the number of which scales linearly with the number of spins, N, compared to an exact solution through diagonalization that, in the case of the Heisenberg model, would require matrices exponentially large in N. This mapping is exact and can be extended to higher dimensional interacting systems as well as to systems with an explicit coupling to the environment.

Thermodynamic Bethe Ansatz equations for two-particle states from the sector proposed by Arutyunov, Suzuki and the author are solved numerically for the Konishi operator descendent up to 't Hooft's coupling λ ≈ 2046. The data obtained is used to analyze the properties of Y-functions and address the issue of the existence of the critical values of the coupling. In addition, we find a new integral representation for the BES dressing phase which substantially reduces the computational time.

A long calculation on the basis of a general inequality by Gaveau and Schulman on decoherence energies is reduced to the application of few basic observations on product structures in a class of time evolutions of product states.