Table of contents

A nonlinear coupled system descriptive of multi-ion electrodiffusion is investigated and all parameters for which the system admits a single-valued general solution are isolated. This is achieved via a method initiated by Painlevé with the application of a test due to Kowalevski and Gambier. The solutions can be obtained explicitly in terms of Painlevé transcendents or elliptic functions.

The presence of the aging phenomenon in the homogeneous cooling state (HCS) of a granular fluid composed of inelastic hard spheres or disks is investigated. As a consequence of the scaling property of the N-particle distribution function, it is obtained that the decay of the normalized two-time correlation functions slows down as the time elapsed since the beginning of the measurement increases. This result is confirmed by molecular dynamics simulations for the particular case of the total energy of the system. The agreement is also quantitative in the low density limit, for which an explicit analytical form of the time correlation function has been derived. Moreover, the reported results provide support for the existence of the HCS as a solution of the N-particle Liouville equation.

We study the spectrum of states, in a plane perpendicular to a uniform magnetic field, for an electron restricted to the lowest Landau level in the presence of randomly distributed, repulsively correlated electric impurities. The lowest energy band in this spectrum would be the lowest Landau level if there were an effective magnetic flux density downshifted by an integer multiple of the impurity density. The downshift is precisely half that for the corresponding electron density in the composite-fermion picture of the fractional quantum Hall regime. Although the work is numerical, the striking result confirms heuristic arguments based on simple adiabatic calculations. This raises a possibility that further development of adiabatic methods might allow deduction of composite-fermion theory for the fractional quantum Hall effect, starting with spin-up electrons restricted to the lowest Landau level and interacting by Coulomb forces.

Vortices are known to play a key role in many important processes in physics and chemistry. Here, we study vortices in connection with the quantum trajectories that can be defined in the framework provided by the de Broglie–Bohm formalism of quantum mechanics. In a previous work, it was shown that the presence of a single moving vortex is enough to induce chaos in these trajectories. Here, this situation is explored in more detail by discussing the relationship between Lyapunov exponents and the parameters characterizing the vortex dynamics. We also consider the issue when more than one vortex exists. In this case, the interaction among them can annihilate or create pairs of vortices with opposite vorticity. This phenomenon is analyzed from a dynamical point of view, showing how the size of the regular regions in phase space grows, as vortices disappear.

Real networks often consist of local units which interact with each other via asymmetric and heterogeneous connections. In this paper, the V-stability problem is investigated for a class of asymmetric weighted coupled networks with nonidentical node dynamics, which includes the unweighted network as a special case. Pinning control is suggested to stabilize such a coupled network. The complicated stabilization problem is reduced to measuring the semi-negative property of the characteristic matrix which embodies not only the network topology, but also the node self-dynamics and the control gains. It is found that network stabilizability depends critically on the second largest eigenvalue of the characteristic matrix. The smaller the second largest eigenvalue is, the more the network is pinning controllable. Numerical simulations of two representative networks composed of non-chaotic systems and chaotic systems, respectively, are shown for illustration and verification.

Dunajski generalization of the second heavenly equation is studied. A dressing scheme applicable to the Dunajski equation is developed; an example of constructing solutions in terms of implicit functions is considered. A Dunajski equation hierarchy is described, and its Lax–Sato form is presented. The Dunajski equation hierarchy is characterized by the conservation of a three-dimensional volume form, in which a spectral variable is taken into account.

We generalize Bochner's theorem for functions of the positive type—theorem 1—to more general integral transforms using the Jost solution of the radial Schrödinger equation. The generalized theorem is theorem 2. We then use Bochner's theorem to obtain an integral representation for the phase shift, shown in theorem 4. In a forthcoming paper, this theorem will be used in inverse scattering theory. The proofs are simple, and make use of well-known theorems of real analysis and Fourier transforms of L¹, L¹∩L², ... functions.

A hidden nonlinear bosonized supersymmetry was revealed recently in the Pöschl–Teller and finite-gap Lamé systems. In spite of the intimate relationship between the two quantum models, the hidden supersymmetry in them displays essential differences. In particular, the kernel of the supercharges of the Pöschl–Teller system, unlike the case of the Lamé equation, includes nonphysical states. By means of the Lamé equation, we clarify the nature of these peculiar states, and show that they encode essential information not only on the original hyperbolic Pöschl–Teller system, but also on its singular hyperbolic and trigonometric modifications, and reflect the intimate relation of the model to a free-particle system.

We use the coordinate Bethe ansatz approach to derive the nested Bethe equations corresponding to the recently found S-matrix for strings in AdS₅ × S⁵, compatible with centrally extended symmetry.

The classical transcendental solutions to the Painlevé III equation are derived from a family of solutions to the anti-self-dual Yang–Mills equation. It is also shown that the affine Weyl group symmetry of P_III is recovered from the symmetry of Yang's equation.

We consider quantum walks on the cycle in the non-stationary case where the 'coin' operation is allowed to change at each time step. We characterize, in algebraic terms, the set of possible state transfers and prove that, as opposed to the stationary case, the associate probability distribution may converge to a uniform distribution among the nodes of the associated graph.

A theoretical scheme to employ the principle of minimal sensitivity for choosing the optimal values of nonlinear parameters is proposed for the multistate Rayleigh–Ritz variational method. Anharmonic oscillators are particularly considered in this paper. Applications of the present scheme to the one-dimensional Morse and two double-well potentials indicate that it provides much more accurate and faster convergent approximations to the exact energy eigenvalues than several schemes existing in the literatures.

Quantum mechanics is based on a series of postulates which lead to a very good description of the microphysical realm but which have, up to now, not been derived from first principles. In the present work, we suggest such a derivation in the framework of the theory of scale relativity. After having analyzed the actual status of the various postulates, rules and principles that underlie the present axiomatic foundation of quantum mechanics (in terms of main postulates, secondary rules and derived 'principles'), we attempt to provide the reader with an exhaustive view of the matter, by both gathering here results which are already available in the literature, and deriving new ones which complete the postulate list.

Open questions from Sarovar and Milburn (2006 J. Phys. A: Math. Gen.39 8487) are answered. Sarovar and Milburn derived a convenient upper bound for the Fisher information of a one-parameter quantum channel. They showed that for quasi-classical models their bound is achievable and they gave a necessary and sufficient condition for positive operator-valued measures (POVMs) attaining this bound. They asked (i) whether their bound is attainable more generally, (ii) whether explicit expressions for optimal POVMs can be derived from the attainability condition. We show that the symmetric logarithmic derivative (SLD) quantum information is less than or equal to the SM bound, i.e., H(θ) ⩽ C_ϒ(θ) and we find conditions for equality. As the Fisher information is less than or equal to the SLD quantum information, i.e., F_M(θ) ⩽ H(θ), we can deduce when equality holds in F_M(θ) ⩽ C_ϒ(θ). Equality does not hold for all channels. As a consequence, the attainability condition cannot be used to test for optimal POVMs for all channels. These results are extended to multi-parameter channels.

Recently, the quantum brachistochrone problem has been discussed in the literature by using non-Hermitian Hamilton operators of different types. Here, it is demonstrated that the passage time is tunable in realistic open quantum systems due to the biorthogonality of the eigenfunctions of the non-Hermitian Hamilton operator. As an example, the numerical results obtained by Bulgakov et al for the transmission through microwave cavities of different shapes are analyzed from the point of view of the brachistochrone problem. The passage time is shortened in the crossover from the weak-coupling to the strong-coupling regime where the resonance states overlap and many branch points (exceptional points) in the complex plane exist. The effect can not be described in the framework of the standard quantum mechanics with the Hermitian Hamilton operator and consideration of S matrix poles.

A microscopic derivation of the master equation for the Jaynes–Cummings model with cavity losses is given, taking into account the terms in the dissipator which vary with frequencies of the order of the vacuum Rabi frequency. Our approach allows us to single out physical contexts wherein the usual phenomenological dissipator turns out to be fully justified and constitutes an extension of our previous analysis (Scala et al2007 Phys. Rev. A 75 013811), where a microscopic derivation was given in the framework of the rotating wave approximation.

An irreducible canonical approach to second-order reducible second-class constraints is given. The procedure is exemplified on gauge-fixed 3-forms.

In the present paper we present in detail a new and improved version of the method of perturbative equivalence between the propagation characteristics of lossless trapezoidal cross-section waveguides and of rectangular cross-section waveguides. The analysis is carried out within the framework of perturbation theory, and to the first order we develop a very simple geometrical argument that allows one to construct for any given waveguide with trapezoidal cross-section an equivalent waveguide with rectangular cross-section. The numerical investigation of our arguments shows excellent agreement between the propagation characteristics of the two equivalent structures.

We study non-Abelian expanding waves that can be radiated from various sources of Yang–Mills fields. We find a new class of exact wave solutions to the Yang–Mills equations. These solutions are constructed for any gauge group with a compact semi-simple Lie algebra and embrace asymmetrical cases of radiations of expanding waves. They can be regarded as a reasonable generalization of wave solutions of the Maxwell electrodynamics. It is of interest to apply the found solutions to detect cosmic sources of Yang–Mills fields. In the case of fields with SU(2) symmetry this could be realized by observing the interaction of such sources' radiation with neutrinos.

We give precise meaning to piecewise constant potentials in non-commutative quantum mechanics. In particular, we discuss the infinite and finite non-commutative spherical wells in two dimensions. Using this, bound states and scattering can be discussed unambiguously. Here we focus on the infinite well and solution for the eigenvalues and eigenfunctions. We find that time reversal symmetry is broken by the non-commutativity. We show that in the commutative and thermodynamic limits the eigenstates and eigenfunctions of the commutative spherical well are recovered and time reversal symmetry is restored.

In this paper, we introduce the condition of θ-locality which can be used as a substitute for microcausality in quantum field theory on noncommutative spacetime. This condition is closely related to the asymptotic commutativity which was previously used in nonlocal QFT. Heuristically, it means that the commutators of observables behave at large spacelike separation like exp(−|x − y|²/θ), where θ is the noncommutativity parameter. The rigorous formulation given in the paper implies averaging fields with suitable test functions. We define a test function space which most closely corresponds to the Moyal ⋆-product and prove that this space is a topological algebra under the star product. As an example, we consider the simplest normal ordered monomial :ϕ⋆ϕ: and show that it obeys the θ-locality condition.