Table of contents

Quantum phase transitions affecting the structure of ground and excited states of integrable systems with the Mexican-hat type potential are shown to be related to a singular torus of classical orbits passing the point of unstable equilibrium. As a specific example, we consider nuclear collective vibrations described by the O(6)–U(5) transitional Hamiltonian of the interacting boson model. While all states with zero values of the O(5) invariant undergo a continuous phase transition when crossing the energy of unstable equilibrium, the other states evolve in an analytic way.

It was shown that quantum analysis constitutes the proper analytic basis for the quantization of Lyapunov exponents in the Heisenberg picture. Differences among various quantizations of Lyapunov exponents are clarified.

Stokes waves on the surface of a layer of an ideal fluid are studied. The nonlinear Schrodinger equation (NSE) for the envelope of the first harmonic and the equation for zero harmonic are extended with allowance for full linear dispersion. To investigate modulational instability (MI) of Stokes waves, we derive a quartic equation for the perturbation frequency without the traditional approximation for the motion of mean current with a group speed on the frequency of fast filling. The interaction of the four roots of this equation is shown to result in the occurrence of MI bands not described by the NSE. The analysis of the obtained expressions demonstrates that the limit kh = 1.363 (where h is the fluid depth and k is the wave number) found by Benjamin and Feir (and also by Whitham and then by Hasimoto and Ono) for the transition between the states of modulationally stable and unstable liquid is valid only in the limiting case of small amplitudes of unperturbed waves and small wave numbers of the perturbation wave.

A lattice gas model is proposed for the A₂ + 2B₂ → 2B₂A reaction system with particle diffusion. In the model, A₂ dissociates in the random dimer-filling mechanism and B₂ dissociation is in the end-on dimer-filling mechanism. A reactive window appears and the system exhibits a continuous phase transition from a reactive state to a covered state with infinitely many absorbing states. When the diffusion of particle A and AB is included, there are still infinitely many absorbing states for the continuous phase transition, but it is found that the critical behaviour changes from the directed percolation (DP) class to the pair contact process with diffusion (PCPD) class.

We calculate the Lax pairs of homogeneous and inhomogeneous one-dimensional time-dependent Gross–Pitaevskii equations with time-dependent scattering length. The inhomogeneity corresponds to linear and quadratic potentials. Our approach introduces a systematic method of searching for the Lax pair corresponding to a given differential equation. We derive known Lax pairs for the Gross–Pitaevskii equation with homogeneous and quadratic potentials and time-dependent scattering length. We also derive new Lax pairs corresponding to a Gross–Pitaevskii equation with a linear potential. Using the resulting Lax pairs, the Darboux transformation can be performed and exact solutions of the Gross–Pitaevskii equation can be obtained for experimentally relevant cases such as solitonic solutions.

We present a theoretical study of the equilibrium ordering in a 3D XY nematic system with quenched random disorder. Within this model, treated with the replica trick and Gaussian variational method, the correlation length is obtained as a function of the local nematic order parameter Q and the effective disorder strength Γ. These results, and ξ ∼ (1/Γ) e^−Γ, clarify what happens in the limiting cases of diminishing Q and Γ, that is near a phase transition of a pure system. In particular, it is found that quenched disorder is irrelevant as Q → 0 and hence does not change the character of the continuous XY nematic–isotropic phase transition. We discuss how these results compare with experiments and simulations.

Introducing sets of constraints, we define new classes of random-matrix ensembles, the constrained Gaussian unitary (CGUE) and the deformed Gaussian unitary (DGUE) ensembles. The latter interpolate between the GUE and the CGUE. We derive a sufficient condition for GUE-type level repulsion to persist in the presence of constraints. For special classes of constraints, we extend this approach to the orthogonal and to the symplectic ensembles. A generalized Fourier theorem relates the spectral properties of the constraining ensembles with those of the constrained ones. We find that in the DGUEs, level repulsion always prevails at a sufficiently short distance and may be lifted only in the limit of strictly enforced constraints.

The complex projection of any n-dimensional quaternionic unitary dynamics defines a one-parameter positive semigroup dynamics. We show that the converse is also true, i.e. that any one-parameter positive semigroup dynamics of complex density matrices with maximal rank can be obtained as the complex projection of suitable quaternionic unitary dynamics.

We explore a nonlocal connection between certain linear and nonlinear ordinary differential equations (ODEs), representing physically important oscillator systems, and identify a class of integrable nonlinear ODEs of any order. We also devise a method to derive explicit general solutions of the nonlinear ODEs. Interestingly, many well-known integrable models can be accommodated into our scheme and our procedure thereby provides further understanding of these models.

The positions of atoms forming a carbon nanotube are usually described by using a system of generators of the symmetry group. Each atomic position corresponds to an element of the set , where n depends on the considered nanotube. We obtain an alternative, rather different description by starting from a three-axes description of the honeycomb lattice. In our mathematical model, which is a factor space defined by an equivalence relation in the set , the neighbours of an atomic position can be described in a simpler way, and the mathematical objects with geometric or physical significance have a simpler and more symmetric form. We present some results concerning the linear representations of the symmetry groups of single-wall carbon nanotubes in order to illustrate the proposed approach.

It is shown that the functional derivatives in density-functional theory (DFT) can be explicitly defined within the domain of electron densities restricted by the electron number, and a constructive definition of such restricted derivatives is suggested. With this definition, Kohn–Sham (KS) equations can be established for an N-electron system without extending the functional domain and introducing a Lagrange multiplier. This may clarify some of the fundamental questions raised by Nesbet (1998 Phys. Rev. A 58 R12). The definition naturally leads to the fact that the KS effective potential is determined only to within an additive constant, thus the KS levels can shift freely and the relation between the highest occupied molecular orbital (HOMO) energy and the ionization potential of the system depends on the choice of the constant. On the other hand, if the domain of functionals is indeed extended beyond the electron number restriction, conclusions depend on whether the extended functionals have unrestricted derivatives or not. It is shown that the ensemble extension of DFT to open systems of mixed states (Perdew et al 1982 Phys. Rev. Lett.49 1691) leads to an energy functional which has no unrestricted derivative at integer electron numbers. Hence after this extension, the relation between the HOMO energy and the ionization potential for an N-electron system is still uncertain. Besides, there are different extensions of the energy functional to a domain of densities unrestricted by the integer electron number, resulting in different unrestricted derivatives and electron systems with different chemical potentials. Even for the exact exchange-correlation potential, there is still an undetermined constant, whether it is a restricted or unrestricted derivative.

We give a general construction for the classical limit of a quantum system defined in terms of generators of an arbitrary compact semisimple Lie algebra, generalizing known results for the and cases. The classical limit depends on the physical problem in question and is determined by the sequence of representations by which it is reached. Only in the simplest cases is it unique. We present explicit formulae useful in determining the classical limit in all important cases.

We consider the fractional generalization of nonholonomic constraints defined by equations with fractional derivatives and provide some examples. The corresponding equations of motion are derived using variational principle. We prove that fractional constraints can be used to describe the evolution of dynamical systems in which some coordinates and velocities are related to velocities through a power-law memory function.

The original Jaynes–Cummings model is described by a Hamiltonian which is Hermitian and exactly solvable. Here, we extend this model by several types of interactions leading to a non-Hermitian operator which does not satisfy the physical condition of spacetime reflection symmetry (PT symmetry). The new Hamiltonians are either exactly solvable admitting an entirely real spectrum or quasi-exactly solvable with a real algebraic part of their spectrum.

We find that the wavefunction of a pair coherent state (or SU(1,1) coherent state) in the entangled state representation is just the eigenfunction of a type of Fokker–Planck differential operator. The relationship between these two states is discussed.

We construct a measure of entanglement by generalizing the quadric polynomial of the Segre variety for general multipartite states. This measure of entanglement works for any pure state and vanishes on multipartite product states. We give explicit expressions for general pure three-partite and four-partite states, and compare our measure of entanglement with the generalized concurrence.

We attempt to clarify certain puzzles concerning state collapse. In open quantum systems decoherence is a necessary consequence of the transfer of information to the outside; we prove an upper bound for the amount of coherence which can survive such a transfer. In large closed systems decoherence is not observed, but we will show that it is usually harmless to assume its occurrence. An independent physical collapse mechanism over and above Schrödinger's equation and the probability interpretation of quantum states is therefore redundant.

In this work we consider a ballistic superconductor/multilayer–ferromagnet/superconductor junction, with interface scattering potential, which can be different in strength. We develop, for an arbitrary number of layers, compact analytic formulae for the Andreev bound state spectrum and the supercurrent. The phase dependence of the supercurrent is obtained from the Andreev amplitudes and results in an extremely simple form for any number of ferromagnetic layers. In the Matsubara summation it enters the denominator as a cos ϕ plus a ϕ-independent term. The reduced formulae are amenable to efficient numerical computation. For a few cases we also give simple equations for the numerical determination of the bound states, obtained with algebraic matrix multiplication and numerical results are discussed.

An approach featuring s-parametrized quasiprobability distribution functions is developed for situations where a circular topology is observed. For such an approach, a suitable set of angle–angular momentum coherent states must be constructed in an appropriate fashion.

We outline a systematic procedure for computing the propagators of quantum fluctuations in spherically symmetric classical backgrounds. As an example, we rederive the propagator for the quantum fluctuations around an instanton background in Yang–Mills theory. The procedure may be applied to several other examples of physical interest, such as fluctuations around vortices, monopoles, skyrmions, etc. Analytic expressions for the propagators may be obtained from analytic solutions of ordinary differential equations. In this construction of the propagator, we present new spherical harmonics for the decomposition o(4) = su(2) ⊕ su(2), which are more convenient for the present example.

The recently investigated Hilbert–Krein and other positivity structures of the superspace are considered in the framework of superdistributions. These tools are applied to problems raised by rigorous supersymmetric quantum field theory.

We obtain lower and upper bounds on the heat kernel and Green functions of the Schrödinger operator in a random Gaussian magnetic field and a fixed scalar potential. We apply the stochastic Feynman–Kac representation, diamagnetic upper bounds and the Jensen inequality for the lower bound. We show that if the covariance of the electromagnetic (vector) potential is increasing at large distances, then the lower bound decreases exponentially fast for large distances and a large time.

We study the effects of a uniform drift on the excitation emanating from an external antenna immersed in a cold modelled plasma. Instead of the well-known approach based on the steady-state regime approximation, the causal and transient evolution of the excited plasma waves is investigated. A comprehensive description of the computational method is proposed. We show that a technique based on the Heaviside direct operational method turns out to be very effective in order to handle the integrodifferential expression for the solution. Making use of a pure analytical calculation, an exact algebraic expression for the plasma response is inferred. The solution takes the form of a combination of Bessel's functions and series of Lommel's functions. Then some specific analysis of the result allow us to gain more understanding on the dynamics of the fast and slow space-charge waves. A special attention is paid to the analysis of the extended singularity of the wave field and the secular behaviour of the fast wave mode at the resonant excitation.

In the recent paper (Pulé J V and Zagrebnov V A 2004 The approximating Hamiltonian method for the imperfect boson gas J. Phys. A: Math. Gen.45 3565–83) the pressure for the imperfect (mean field) boson gas is derived by a method based on the approximating Hamiltonian argument. We give some comments about this derivation.