Table of contents

In this letter, we introduce a new generalized linearizing transformation (GLT) for second-order nonlinear ordinary differential equations (SNODEs). The well-known invertible point (IPT) and non-point transformations (NPT) can be derived as sub-cases of the GLT. A wider class of nonlinear ODEs that cannot be linearized through NPT and IPT can be linearized by this GLT. We also illustrate how to construct GLTs and to identify the form of the linearizable equations and propose a procedure to derive the general solution from this GLT for the SNODEs. We demonstrate the theory with two examples which are of contemporary interest.

The difficulties arising in the investigation of finite-size scaling in d-dimensional O(n) systems with strong anisotropy and/or long-range interaction, decaying with the interparticle distance r as r^−d−σ(0 < σ ⩽ 2), are discussed. Some integral representations aiming at the simplification of the investigations are presented for the classical and quantum lattice sums that take place in the theory. Special attention is paid to a more general form allowing to treat both cases on an equal footing and in addition cases with strong anisotropic interactions and different geometries. The analysis is simplified further by expressing this general form in terms of a generalization of the Mittag–Leffler special functions. This turned out to be very useful for the extraction of asymptotic finite-size behaviours of the thermodynamic functions.

In the present paper, we revisit nonlinearity management of the time-periodic nonlinear Schrödinger equation and the related averaging procedure. By means of rigorous estimates, we show that the averaged nonlinear Schrödinger equation does not blow up in the higher dimensional case so long as the corresponding solution remains smooth. In particular, we show that the H¹ norm remains bounded, in contrast with the usual blow-up mechanism for the focusing Schrödinger equation. This conclusion agrees with earlier works in the case of strong nonlinearity management but contradicts those in the case of weak nonlinearity management. The apparent discrepancy is explained by the divergence of the averaging procedure in the limit of weak nonlinearity management.

The motion of a classical Hamiltonian oscillator, driven by a weak random force, is examined by means of a deterministic approach. We design the specific Poincaré map to find domains of finite-time stability in phase space. The trajectories belonging to these domains remain stable by Lyapunov criteria for the period of mapping T₀ at least. We derive the lower border for the time of phase correlations between close trajectories. It is found that the lifetime of some stable domains significantly exceeds the correlation time of the external force. The randomly-driven Morse oscillator is used as an example.

In a previous work (Bouferguene 2005 J. Phys. A: Math. Gen.38 3923), we have shown that in the framework of the Gaussian integral transform, multi-centre integrals over Slater type functions can be evaluated to an acceptable accuracy using a tailored Gauss quadrature in which the weight function has the form W(σ, τ; z) = z^νexp(−σz − τ/z). To be considered a solution worth implementing within a software for routine use in ab initio molecular simulations, the method must also prove to be at least as efficient as those methods previously published in the literature. Two major results are provided in this paper. Firstly, an improvement of the procedure used to generate the roots and weights of the Gauss–Bessel quadrature is proposed. Secondly, a computational cost analysis of the present method and the (Safouhi 2001 J. Phys. A: Math. Gen.34 2801) based approach are compared, hence proving the equivalence of the two from a complexity point of view.

Some types of coupled Korteweg de-Vries (KdV) equations are derived from a two-layer fluid system. In the derivation procedure, an unreasonable y-average trick (usually adopted in the literature) is removed. The derived models are classified by means of the Painlevé test. Three types of τ-function and multiple soliton solutions of the models are explicitly given via the exact solutions of the usual KdV equation. It is also discovered that a non-Painlevé integrable coupled KdV system can have multiple soliton solutions.

In the framework of a recently proposed topological approach to phase transitions, some sufficient conditions ensuring the presence of the spontaneous breaking of a symmetry and of a symmetry-breaking phase transition are introduced and discussed. A very simple model, which we refer to as the hypercubic model, is introduced and solved. The main purpose of this model is that of illustrating the content of the sufficient conditions, but it is interesting also in itself due to its simplicity. Then some mean-field models already known in the literature are discussed in the light of the sufficient conditions introduced here.

The analytic solutions of the one-dimensional Schrödinger equation for the trigonometric Rosen–Morse potential reported in the literature rely upon the Jacobi polynomials with complex indices and complex arguments. We first draw attention to the fact that the complex Jacobi polynomials have non-trivial orthogonality properties which make them uncomfortable for physics applications. Instead we here solve the above equation in terms of real orthogonal polynomials. The new solutions are used in the construction of the quantum-mechanical superpotential.

In a variational formulation of the interaction of a point particle with the spin 0 Schrödinger field in which there is no back-reaction of the particle on the wave, it is shown that the condition that (a certain subset of the) components of the current and energy–momentum complexes of the composite take their quantum values fixes the particle law of motion as that given by de Broglie and Bohm. No appeal to statistics is made.

In density functional theory, the single-particle kinetic energy is still not known in an orbital-free form. The natural tool to calculate such kinetic energy for a given one-body potential V is the Dirac density matrix γ. Here, by first taking solvable one-dimensional examples, and in particular the sech²(x) potential, forms for the Dirac density matrix are exhibited in terms of the potential.This in turn generates the Legendre transform of the single-particle kinetic energy. Finally, one-dimensional perturbation theory for the kinetic energy density, summed to all orders in V, is presented in the appendix.

Damping of multiphoton Rabi oscillations due to a finite lifetime of photons is investigated using an algebraic method of the solution of the master equation. The time dependence of the probability of electron transitions between atomic levels is evaluated. In a regime of weak atom–photon coupling (a coupling constant smaller than the photon decay rate), the spontaneous emission of many photons from a single electron transition due to the interaction with a resonant light is found. Thus, a multiphoton counterpart of the Purcell effect is predicted.

We address the problem of the construction of quantum walks on Cayley graphs. Our main motivation is the relationship between quantum algorithms and quantum walks. In particular, we discuss the choice of the dimension of the local Hilbert space and consider various classes of graphs on which the structure of quantum walks may differ. We completely characterize quantum walks on free groups and present partial results on more general cases. Some examples are given including a family of quantum walks on the hypercube involving a Clifford algebra.

The weak energy for a quantum state evolving relative to its fixed initial state (pointed weak energy) in a Hilbert space of arbitrary dimension is defined and aspects of the classical mechanical, transformational and geometric relationships between this quantity and the motion of the associated quantum state are studied in a two-dimensional configuration space with Pancharatnam (P) phase and Fubini-Study (FS) metric distance as generalized coordinates (PFS configuration space). It is shown that when expressed in these coordinates: (1) the evolution of the quantum state is described by an equation of motion for an associated correlation amplitude (pointed correlation amplitude) in which the dynamics are induced by the interaction of the pointed weak energy with this correlation amplitude; (2) the pointed weak energy defines a 1-form with the transformational properties of a U(1) gauge potential (pointed weak energy gauge potential) which produces infinitesimal changes in the pointed correlation amplitude via its interaction with the amplitude; (3) the pointed correlation amplitude is defined by an exponential function of a line integral of the pointed weak energy gauge potential; (4) pointed weak energy gauge transformations exist that are equivalent to local U(1) gauge transformations of quantum states and—when examined within the context of the pointed probability current and pointed weak energy stationary action principles—offer an interpretation of gauge invariance analogous to that found in classical mechanics; and (5) integral invariants exist which relate line integrals of the weak energy gauge potential to geometric phase and the Aharonov–Anandan connection 1-form in the associated Hilbert space. States which irreversibly decay and Grover's quantum search algorithm serve as simple illustrations of the theory.

The separability modulus ℓ(ρ) of a state ρ of an arbitrary finite composite quantum system is the largest t in [0, 1] such that t ⋅ ρ + (1 − t) ⋅ τ is separable, where τ is the normalized trace. The basic properties of ℓ, introduced by Vidal and Tarrach in another guise, are briefly established. With these properties, we obtain conditions on the spectrum of a state which imply that it is separable. As a consequence, we show that for any Hamiltonian H the thermal equilibrium states e^−H/T/Tr(e^−H/T) are separable if T is large enough. Also, for F a unitarily invariant, convex continuous real-valued function on states, for which F(ρ) > F(τ) whenever ρ ≠ τ, there is a critical C_F such that F(ρ) ⩽ C_F implies that ρ is separable, and for each possible c > C_F there are entangled states ϕ with F(ϕ) = c. This class includes all strictly convex unitarily invariant continuous functions, and also every non-trivial partial eigenvalue-sum. Some C_F are computed. General upper and lower bounds for C_F are given, and then improved for bipartite systems.

In the biaxial nanospin system, quantum tunnelling of the magnetization vector oscillates with increasing magnetic field or temperature. When the number of instanons is equal to the number of anti-instantons, the annihilation of instanton–anti-instanton pair results in a vanished Berry curvature, it leads to the topological quenching point. When the number of instantons and anti-instantons is not equal, berry curvature is nonzero, this suggests that there are extra instantons that revive the tunnelling splitting. So the generation (annihilation) of instanton induced the revival (quenching) of tunnel splitting. This generation or annihilation is accompanied by different phase transitions which can be distinguished by . When the system slides from the phase of H/H_c < 1 − K₂/K₁ into the phase of H/H_c > 1 − K₂/K₁, an instanton will be generated (annihilated), which revives (quenches) the quantum tunnelling. This conclusion also holds when phase transition occurs in the opposite direction, i.e., from H/H_c > 1 − K₂/K₁ to H/H_c < 1 − K₂/K₁. So there is a phase transition that occurs alternatively in the magnetic molecule system when the topological quenching of the tunnel splitting occurs quasi-periodically.

A functional renormalization group approach is applied to a single quantum particle with emphasis on quantum tunnelling through a barrier. We include both an effective potential and a wavefunction renormalization and compare the effects of soft cut-off, hard cut-off and Schwinger proper time regulators. Numerical procedures and results are studied in detail.

The relativistic framework with its symmetries offers a natural definition for the internal time of classical (non-quantum) physical systems as a Lorentz-invariant observable. The internal-time observable, measuring the system's aging or internal evolution, is identified with the proper time of the system derived from its centre-of-mass (CM) coordinate. For its definition as an observable it is required that the system be symmetric not only under Lorentz–Poincaré transformations but also under uniform scaling, with the associated existence of a dilatation function D, and yet that D be a varying—not conserved—quantity. Two alternative definitions are discussed, and it is found that in order to maintain simultaneity of the CM time with the events that define it, it is necessary to split the dilatation function into a CM part and an internal part.

Quantum fluctuations of massless scalar fields represented by quantum fluctuations of the quasiparticle vacuum in a zero-temperature dilute Bose–Einstein condensate may well provide the first experimental arena for measuring the Casimir force of a field other than the electromagnetic field. This would constitute a real Casimir force measurement—due to quantum fluctuations—in contrast to thermal fluctuation effects. We develop a multidimensional cut-off technique for calculating the Casimir energy of massless scalar fields in d-dimensional rectangular spaces with q large dimensions and d − q dimensions of length L and generalize the technique to arbitrary lengths. We explicitly evaluate the multidimensional remainder and express it in a form that converges exponentially fast. Together with the compact analytical formulae we derive, the numerical results are exact and easy to obtain. Most importantly, we show that the division between analytical and remainder is not arbitrary but has a natural physical interpretation. The analytical part can be viewed as the sum of individual parallel plate energies and the remainder as an interaction energy. In a separate procedure, via results from number theory, we express some odd-dimensional homogeneous Epstein zeta functions as products of one-dimensional sums plus a tiny remainder and calculate from them the Casimir energy via zeta function regularization.