Table of contents

A class of Fredholm integral equations of the second kind is studied, with kernel separable outside the basic interval (a, b). Using theorems of matrix algebra, the solution for x outside (a, b) is found in terms of the Fredholm determinants in a simple and compact form. As a particular case, the quantum propagator for one-dimensional problems is obtained.

In the last decades, renormalization group (RG) ideas have been applied to describe universal properties of different routes to chaos (quasi-periodic, period doubling or tripling, Siegel disc boundaries, etc). Each of the RG theories leads to universal scaling exponents which are related to the action of certain RG operators. The goal of this announcement is to show that there is a principle that organizes many of these scaling exponents. We give numerical evidence that the exponents of different routes to chaos satisfy approximately some arithmetic relations. These relations are determined by combinatorial properties of the route and become exact in an appropriate limit.

We review recent results concerning the renormalization group (RG) transformation of Dyson's hierarchical model (HM). This model can be seen as an approximation of a scalar field theory on a lattice. We introduce the HM and show that its large group of symmetry simplifies drastically the blockspinning procedure. Several equivalent forms of the recursion formula are presented with unified notations. Rigourous and numerical results concerning the recursion formula are summarized. It is pointed out that the recursion formula of the HM is inequivalent to both Wilson's approximate recursion formula and Polchinski's equation in the local potential approximation (despite the very small difference with the exponents of the latter). We draw a comparison between the RG of the HM and functional RG equations in the local potential approximation. The construction of the linear and nonlinear scaling variables is discussed in an operational way. We describe the calculation of non-universal critical amplitudes in terms of the scaling variables of two fixed points. This question appears as a problem of interpolation between these fixed points. Universal amplitude ratios are calculated. We discuss the large-N limit and the complex singularities of the critical potential calculable in this limit. The interpolation between the HM and more conventional lattice models is presented as a symmetry breaking problem. We briefly introduce models with an approximate supersymmetry. One important goal of this review is to present a configuration space counterpart, suitable for lattice formulations, of functional RG equations formulated in momentum space (often called exact RG equations and abbreviated ERGE).

We have considered the persistence of unvisited sites of a lattice, i.e., the probability S(t) that a site remains unvisited till time t in the presence of mutually repulsive random walkers in one dimension. The dynamics of this system has direct correspondence to that of the domain walls in a certain system of Ising spins where the number of domain walls becomes fixed following a zero-temperature quench. Here we get the result that S(t) ∝ exp(−αt^β) where β is close to 0.5 and α a function of the density of the walkers ρ. The fraction of persistent sites in the presence of independent walkers of density ρ' is known to be . We show that a mapping of the interacting walkers' problem to the independent walkers' problem is possible with ρ' = ρ/(1 − ρ) provided ρ' and ρ are small. We also discuss some other intricate results obtained in the interacting walkers' case.

An analytical approximation method to clarify the symmetry-breaking stationary solutions to the Gross–Pitaevskii equation with symmetric double-well external potential is presented. By this method, we can understand that each symmetry-breaking solution bifurcates from a symmetry-preserving solution as one varies the coupling constant of the nonlinear interaction, and predict the bifurcation point. Also, the method gives a good approximation to the energy eigenvalues of the lower states, symmetry-breaking as well as symmetry-preserving, as long as the nonlinear interaction is not too strong.

This paper considers two new multifractional stochastic processes, namely the Weyl multifractional Ornstein–Uhlenbeck process and the Riemann–Liouville multifractional Ornstein–Uhlenbeck process. Basic properties of these processes such as locally self-similar property and Hausdorff dimension are studied. The relationship between the multifractional Ornstein–Uhlenbeck processes and the corresponding multifractional Brownian motions is established.

We study the transport of an overdamped Brownian particle moving in an asymmetric two-dimensional periodic potential in the presence of an asymmetric unbiased external force. Based on the effective potential approach, the two-dimensional system is simplified to the one-dimensional system. The expression of the net current is obtained at a quasi-steady-state limit. The competitions among the asymmetric parameter between the potentials, the coupling parameter of the potential and the temporally asymmetric parameter of the external force lead to current reversals. There may be three optimized values of temperature at which the current takes its extremum value.

We present an operator approach to deriving Mehler's formula and the Rogers formula for the bivariate Rogers–Szegö polynomials h_n(x, y|q). The proof of Mehler's formula can be considered as a new approach to the nonsymmetric Poisson kernel formula for the continuous big q-Hermite polynomials H_n(x; a|q) due to Askey, Rahman and Suslov. Mehler's formula for h_n(x, y|q) involves a ₃ϕ₂ sum and the Rogers formula involves a ₂ϕ₁ sum. The proofs of these results are based on parameter augmentation with respect to the q-exponential operator and the homogeneous q-shift operator in two variables. By extending recent results on the Rogers–Szegö polynomials h_n(x|q) due to Hou, Lascoux and Mu, we obtain another Rogers-type formula for h_n(x, y|q). Finally, we give a change of base formula for H_n(x; a|q) which can be used to evaluate some integrals by using the Askey–Wilson integral.

Given a unicursal quartic in a plane and four bi-tangential straight lines, there exist a pair of conics four times tangential to it. Choosing an arbitrary point K₀ outside the plane, the lines and either one of the conics determine four planes and a cone, and a quartic surface can be found that stays tangential to them and presents 15 conic point singularities, including K₀. We introduce a one-parameter family of symmetric tensors which are polynomial functions of degree 7 of the Cartesian coordinates. Each of them determines two families of curves on the surface, which are the null curves of the associated metric and are the integral curves of a differential system which turns out to possess the Painlevé property and to be integrable by quadratures. Differential systems of that form have been found (Gaffet 2006 J. Phys. A: Math. Gen.39 99) to represent the rotating motion with precession of gas clouds expanding into a vacuum.

The Krazer–Prym theorem extending the Poisson summation formula to oblique lattices is proved as a direct consequence of translational symmetry. A generalized Ewald approach based on this theorem is reproduced. The two other famous approaches proposed by Nijboer and De Wette, and by Harris and Monkhorst, are derived in the same fashion. The absence of any uniform contribution to bulk electrostatic potentials in each of those treatments is emphasized as a property of locally neutral systems subject to translational symmetry. It is found that an analytic evaluation of the Ewald variable parameter naturally follows from the treatment of the Coulomb lattice characteristic in terms of the Krazer–Prym theorem. The extension of those characteristics to off-lattice points is also discussed.

We present a simple approach for finding an N-soliton solution and the corresponding Jost solutions of the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions. Soliton perturbation theory based on the inverse scattering transform method is developed. As an application of the present theory we consider the action of the diffusive-type perturbation on a single bright/dark soliton.

The study of solitary wave solutions is of prime significance for the nonlinear Schrödinger equation with higher order dispersion and/or higher degree nonlinearities in nonlinear physical systems. We derive the discrete cubic–quintic nonlinear Schrödinger equation from a Hamiltonian using different Poisson brackets. By using the extended Jacobian elliptic function approach, we investigate the abundant exact stationary solitons and periodic waves solution of this equation. These solutions include, Jacobian periodic solutions, alternating phase Jacobi periodic solution, kink and bubble soliton solutions, alternating phase kink soliton solution and alternating phase bubble soliton solution, provided that coefficients are bound by special relation. And then with the aid of symbolic computation, we present in explicit form these solutions. The stability of bubble and kink soliton as well as alternating kink and alternating bubble soliton are also investigated.

In this paper we develop a dressing method for constructing and solving some classes of matrix quasilinear partial differential equations (PDEs) in arbitrary dimensions. This method is based on a homogeneous integral equation with a nontrivial kernel, which allows one to reduce the nonlinear PDEs to systems of non-differential (algebraic or transcendental) equations for the unknown fields. In the simplest examples, the above dressing scheme captures matrix equations integrated by the characteristics method and nonlinear PDEs associated with matrix Hopf–Cole transformations.

A large number of regular and some singular bound state properties of the ground 1¹S(L = 0)-states of the positronium Ps⁻ and hydrogen ^∞H⁻ negative ions are determined to a very high numerical accuracy. The highly accurate variational wavefunctions are constructed with the use of exponential basis functions written in the three-body perimetric coordinates. In particular, the total energy of the ground state of the Ps⁻ ion determined in our calculations is E = −0.262 005 070 232 980 107 770 3745 au, while the analogous ground state energy for the ^∞H⁻ ion is E = −0.527 751 016 544 377 196 589 733 au. These values are the best-to-date variational energies obtained for these systems.

General analytic energy bounds are derived for N-boson systems governed by semirelativistic Hamiltonians of the form where V(r) is a static attractive pair potential. A translation-invariant model Hamiltonian H_c is constructed. We conjecture that ⟨H⟩ ⩾ ⟨H_c⟩ generally, and we prove this for N = 3, and for N = 4 when m = 0. The conjecture is also valid generally for the harmonic oscillator and in the nonrelativistic large-m limit. This formulation allows reductions to scaled 3- or 4-body problems, whose spectral bottoms provide energy lower bounds. The example of the ultrarelativistic linear potential is studied in detail and explicit upper- and lower-bound formulae are derived and compared with earlier bounds.

A Fourier transform in a multimode system is studied, using the Bargmann representation. The growth of a Bargmann function is shown to be related to the second-order correlation of the corresponding state. Both the total growth and the total second-order correlation remain unchanged under the Fourier transform. Examples with coherent states, squeezed states and Mittag–Leffler states are discussed.

We have studied the nonlinear model for interaction of N ions in a trap with a laser beam. By applying two successive unitary transformations, this nonlinear quantum system can be transformed into the multi-levels Jayness–Cummings model. In the end, we can obtain an analytical solution to the two-ion system without performing the Lamb–Dicke approximation, and observe the super-revival phenomenon.

If the fundamental quarks of QCD are replaced by massless adjoint quarks, the pattern of the chiral symmetry breaking drastically changes compared to the standard one. It becomes SU(N_f) → SO(N_f). While for N_f = 2, the chiral Lagrangian describing the 'pion' dynamics is well known, this is not the case at N_f > 2. We outline a general strategy for deriving chiral Lagrangians for the coset spaces and study in detail the case of N_f = k = 3. We obtain two- and four-derivatives terms in the chiral Lagrangian on the coset space = SU(3)/SO(3), as well as the Wess–Zumino–Novikov–Witten term, in terms of an explicit parameterization of the quotient manifold. Then we discuss stable topological solitons supported by this Lagrangian. Aspects of relevant topological considerations scattered in the literature are reviewed. The same analysis applies to SO(N) gauge theories with N_f Weyl flavours in the vector representation.

We show that the mode corresponding to a point of the essential spectrum of the electromagnetic scattering operator is a vector-valued distribution representing the square root of the three-dimensional Dirac's delta function. An explicit expression for this singular mode in terms of the Weyl sequence is provided and analysed. The essential resonance occurs if the permittivity of an object gets close to zero, which is often the case in plasmas and negative-permittivity metamaterials. Such resonance would lead to a perfect localization (confinement) of the electromagnetic field. Simultaneously, however, a portion of electromagnetic energy is removed from the Hilbert space and therefore the whole process may be viewed as absorption.