Table of contents

In this letter, we exactly solve the MFPT problem for a one-dimensional random walk with random step size where there are two fixed reflecting boundaries and a fluctuating absorbing boundary in between. We discuss the consequences of these results in the context of site-specific DNA–protein and DNA–probe interactions.

The one-dimensional Holstein model of spinless fermions interacting with dispersionless phonons is solved by using a Bethe ansatz in analogue to that for the one-dimensional spinless Fermi–Hubbard model. Excitation energies and the corresponding wavefunctions of the model are determined by a set of partial differential equations. It is shown that the model is, at least, quasi-exactly solvable for the two-site case, when the phonon frequency, the electron–phonon coupling strength and the hopping integral satisfy certain relations. As examples, some quasi-exact solutions of the model for the two-site case are derived.

If a system described in terms of empty and filled states (○ and •, respectively) interacts with particles M via the following reactions: (with α being a real number), then it may be described in terms of fractional statistics. On the other hand, provided that the states are identified with adsorption sites, this scheme may be used, according to the value assigned to α, for the description of adsorption–desorption phenomena such as the inhibition of empty sites by the adsorption of polyatomic molecules, multilayer adsorption, and the branched growth of dendrimers. Not only does the fractional statistical treatment reproduce the well-known Langmuir and Brunauer–Emmett–Teller isotherms for α = 0 and α = 1, but also it provides the adsorption isotherms for all other values of α. It is also shown that considered as a function of the parameter α the equilibrium isotherm undergoes a catastrophe at α = 1.

Taking the isotropic limit Δ → 1 in a recent representation theoretic construction of Baxter's Q-operators for the XXZ model with quasi-periodic boundary conditions we obtain new results for the XXX model. We show that quasi-periodic boundary conditions are needed to ensure convergence of the Q-operator construction and derive a quantum Wronskian relation which implies two different sets of Bethe ansatz equations, one above, the other below the 'equator' of total spin S^z = 0. We discuss the limit to periodic boundary conditions at the end and explain how this construction relates to the trace functional introduced by Boos et al in the context of correlation functions on the infinite lattice. We also identify a special subclass of solutions to the quantum Wronskian and numerically verify them up to spin chains of ten sites. This special type of solutions might persist for longer chains.

The critical two-terminal conductance g_c and the spatial fluctuations of critical eigenstates are investigated for a disordered two-dimensional model of non-interacting electrons subject to spin-orbit scattering (Ando model). For square samples, we verify numerically the relation σ_c = 1/[2π(2 − D(1))]e²/h between critical conductivity σ_c = g_c = (1.42 ± 0.005)e²/h and the fractal information dimension of the electron wave function, D(1) = 1.889 ± 0.001. Through a detailed numerical scaling analysis of the two-terminal conductance we also estimate the critical exponent ν = 2.80 ± 0.04 that governs the quantum phase transition.

We investigate one-dimensional driven diffusive systems where particles may also be created and annihilated in the bulk with sufficiently small rate. In an open geometry, i.e., coupled to particle reservoirs at the two ends, these systems can exhibit ergodicity breaking in the thermodynamic limit. The triggering mechanism is the random motion of a shock in an effective potential. Based on this physical picture we provide a simple condition for the existence of a non-ergodic phase in the phase diagram of such systems. In the thermodynamic limit this phase exhibits two or more stationary states. However, for finite systems transitions between these states are possible. It is shown that the mean lifetime of such a metastable state is exponentially large in the system size. As an example the ASEP with the A ∅ A ⇌ AAA reaction kinetics is analysed in detail. We present a detailed discussion of the phase diagram of this particular model which indeed exhibits a phase with broken ergodicity. We measure the lifetime of the metastable states with a Monte Carlo simulation in order to confirm our analytical findings.

We introduce a minimal extended evolving model for small-world networks which is controlled by a parameter. In this model, the network growth is determined by the attachment of new nodes to already existing nodes that are geographically close. We analyse several topological properties for our model both analytically and by numerical simulations. The resulting network shows some important characteristics of real-life networks such as small-world effect and high clustering.

For the case when all discrete spectral parameters are purely imaginary, an explicit N-soliton solution for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions, consisting of arbitrary number of pure bright and/or dark solitons, is derived. Shifts of soliton positions due to collisions between solitons are analytically obtained, which are irrespective of the bright or dark characters of the participating solitons. Typical collisions between solitons are graphically shown.

A solution of large multivalent ions in contact with a planar charged surface is considered theoretically. Using a simple three-state model for the orientation of the multivalent ions in the gradient of an electric field and applying the methods of statistical physics, we show that the internal charge distribution within a single multivalent ion with the charges located at well-separated positions may lead to the orientational ordering of multivalent ions near the charged planar surface.

The general variable separated approach is successfully extended to the (2+1)-dimensional physical model and we obtain a universal formula involving arbitrary number of variable separated functions. Because of the existence of the arbitrary functions in the universal formula, many new types of structures of periodic waves, for example, the periodic–periodic interaction waves, periodic–kink interaction waves, periodic–peakon interaction waves, periodic–compacton interaction waves, periodic–foldon interaction waves, etc, are investigated both analytically and graphically. Some novel features or interesting behaviours are revealed.

The tetrahedron equation is a three-dimensional generalization of the Yang–Baxter equation. Its solutions define integrable three-dimensional lattice models of statistical mechanics and quantum field theory. Their integrability is not related to the size of the lattice, therefore the same solution of the tetrahedron equation defines different integrable models for different finite periodic cubic lattices. Obviously, any such three-dimensional model can be viewed as a two-dimensional integrable model on a square lattice, where the additional third dimension is treated as an internal degree of freedom. Therefore every solution of the tetrahedron equation provides an infinite sequence of integrable 2d models differing by the size of this 'hidden third dimension'. In this paper, we construct a new solution of the tetrahedron equation, which provides in this way the two-dimensional solvable models related to finite-dimensional highest weight representations for all quantum affine algebra , where the rank n coincides with the size of the hidden dimension. These models are related to an anisotropic deformation of the sl(n)-invariant Heisenberg magnets. They were extensively studied for a long time, but the hidden 3d structure was hitherto unknown. Our results lead to a remarkable exact 'rank-size' duality relation for the nested Bethe Ansatz solution for these models. Note also that the above solution of the tetrahedron equation arises in the quantization of the 'resonant three-wave scattering' model, which is a well-known integrable classical system in 2 + 1 dimensions.

We obtain a lowering operator for multiple orthogonal polynomials having orthogonality conditions with respect to classical weights. These multiple orthogonal polynomials are generalizations of the classical orthogonal polynomials. Combining the lowering operator with the raising operators, which have been obtained earlier in the literature, we then also obtain a linear differential equation of order r + 1.

In this paper we review the recent results about the flag curvature of invariant Randers metrics on homogeneous manifolds. By using some counterexamples we show that the formula obtained for the flag curvature of these metrics is incorrect. Then we give an explicit formula for the flag curvature of invariant Randers metrics on naturally reductive homogeneous manifolds (G/H, g), where the Randers metric is induced by the invariant Riemannian metric g and an invariant vector field which is parallel with g.

We are concerned with a certain class of Schlömilch series that arise naturally in the study of diffraction problems when the scatterer is a semi-infinite periodic structure. By combining new results derived from integral representations and the Poisson summation formula with known identities, we obtain expressions which enable the series to be computed accurately and efficiently. Many of the technical details of the derivations are omitted; they can, however, be obtained from Linton 2005 Schlömilch series that arise in diffraction theory and their efficient computation Technical Report Loughborough University available online at http://www-staff.lboro.ac.uk/~macml1/schlomilch-techreport.pdf.

The ordinary Bondi–Metzner–Sachs (BMS) group B is the common asymptotic symmetry group of all radiating, asymptotically flat, Lorentzian spacetimes. As such, B is the best candidate for the universal symmetry group of general relativity. However, in studying quantum gravity, spacetimes with signatures other than the usual Lorentzian one and complex spacetimes are frequently considered. Generalizations of B appropriate to these other signatures have been defined earlier. In particular, the generalization B(2, 2) appropriate to the ultrahyperbolic signature (+, +, −, −) has been described in detail, and the study of its irreducible unitary representations (IRs) of B(2, 2) has been initiated. The infinite little groups have been given explicitly, but the finite little groups have only been partially described. This study is completed by describing in detail the finite little groups and by giving all the necessary information in order to construct the IRs of B(2, 2) in all cases.

In this paper, we first present the Grammian determinant solutions to the nonisospectral Kadomtsev–Petviashvili (KP) equation. Then, by using the Pfaffianization procedure of Hirota and Ohta, an integrable coupled system is generated. Moreover, Gramm-type Pfaffian solutions to the Pfaffianized system are proposed.

We address the problem of transmitting states belonging to finite dimensional Hilbert space through a quantum channel associated with a larger (even infinite dimensional) Hilbert space.

Given two spherically symmetric and short-range potentials V₀ and V₁ for which the radial Schrödinger equation can be solved explicitly at zero energy, we show how to construct a new potential V for which the radial equation can again be solved explicitly at zero energy. The new potential and its corresponding wavefunction are given explicitly in terms of V₀ and V₁, and their corresponding wavefunctions φ₀ and φ₁. V₀ must be such that it sustains no bound states (either repulsive, or attractive but weak). However, V₁ can sustain any (finite) number of bound states. The new potential V has the same number of bound states, by construction, but the corresponding (negative) energies are, of course, different. Once this is achieved, one can start then from V₀ and V, and construct a new potential for which the radial equation is again solvable explicitly. And the process can be repeated indefinitely. We exhibit first the construction, and the proof of its validity, for regular short-range potentials, i.e. those for which rV₀(r) and rV₁(r) are L¹ at the origin. It is then seen that the construction extends automatically to potentials which are singular at r = 0. It can also be extended to V₀ long range (Coulomb, etc). We finally give several explicit examples.

In this paper we consider a quantized single-mode field interacting with two two-level atoms. For the radiation field of this system we investigate the occurrence of the revival-collapse phenomenon in the evolution of the Wigner (W) function at the phase-space origin and of the quadrature squeezing. We show under certain conditions that the higher-order quadrature squeezing and the W function can provide complete information on the total atomic inversion. Moreover, we develop the notion of the geometric atomic inversion and show that it can be connected by the W function and the quadrature squeezing.

In this paper, we study the symmetry known (Landau and Lifshits 1976 Course of Theoretical Physics vol 1: Mechanics (Oxford: Pergamon)) as mechanical similarity (LMS) and present for any monomial potential. We analyse it in the framework of the Koopman–von Neumann formulation of classical mechanics and prove that in this framework the LMS can be given a canonical implementation. We also show that the LMS is a generalization of the scale symmetry which is present only for the inverse square and a few other potentials. Finally, we study the main obstructions which one encounters in implementing the LMS at the quantum-mechanical level.

There has been renewed interest in the exploitation of Barta's configuration space theorem (BCST) (Barta 1937 C. R. Acad. Sci. Paris204 472) which bounds the ground-state energy, , by using any Φ lying within the space of positive, bounded, and sufficiently smooth functions, . Mouchet's (Mouchet 2005 J. Phys. A: Math. Gen.38 1039) BCST analysis is based on gradient optimization (GO). However, it overlooks significant difficulties: (i) appearance of multi-extrema; (ii) inefficiency of GO for stiff (singular perturbation/strong coupling) problems; (iii) the nonexistence of a systematic procedure for arbitrarily improving the bounds within . These deficiencies can be corrected by transforming BCST into a moments' representation equivalent, and exploiting a generalization of the eigenvalue moment method (EMM), within the context of the well-known generalized eigenvalue problem (GEP), as developed here. EMM is an alternative eigenenergy bounding, variational procedure, overlooked by Mouchet, which also exploits the positivity of the desired physical solution. Furthermore, it is applicable to Hermitian and non-Hermitian systems with complex-number quantization parameters (Handy and Bessis 1985 Phys. Rev. Lett.55 931, Handy et al 1988 Phys. Rev. Lett.60 253, Handy 2001 J. Phys. A: Math. Gen.34 5065, Handy et al2002 J. Phys. A: Math. Gen.35 6359). Our analysis exploits various quasi-convexity/concavity theorems common to the GEP representation. We outline the general theory, and present some illustrative examples.

A close inspection on the three-dimensional hydrogen atom Hamiltonian revealed formal eigenvectors often discarded in the literature. Although not in its domain, such eigenvectors belong to the Hilbert space, and so their time evolution is well defined. They are then related to the one-dimensional and two-dimensional hydrogen atoms and it is numerically found that they have continuous components, so that ionization can take place.

The classical electromagnetic interaction of a point charge and a magnet is discussed by first calculating the interaction of a point charge with a simple model magnetic moment and then suggesting a multiparticle limit. The Darwin–Lagrangian is used to analyse the electromagnetic behaviour of the model magnetic moment (composed of two oppositely charged particles of different masses in an initially circular Coulomb orbit) interacting with a passing point charge. Considerations of force, energy, momentum and centre of energy are treated through second order in 1/c. The changing magnetic moment is found to put a force back on a passing charge; this force is of order 1/c² and depends upon the magnitude of the magnetic moment. The limit of a many-particle magnet arranged as a toroid is discussed. It is suggested that in the multiparticle limit, the electric fields of the passing charge are screened out of the body of the magnet while the magnetic fields of the passing charge penetrate into the body of the magnet. This is consistent with our understanding of the penetration of electromagnetic velocity fields into ohmic conductors. The proposed multiparticle limit is consistent with the conservation laws for energy and momentum, as well as constant motion of the centre of energy, and Newton's third law for the net Lorentz forces on the magnet and on the point charge. The work corresponds to a classical electromagnetic analysis of the interaction which is basic to understanding the controversy over the Aharonov–Bohm and Aharonov–Casher phase shifts and represents a refutation of the suggestions of Aharonov, Pearle and Vaidman.

The interaction between kink and radiation in nonlinear one-dimensional real scalar field is investigated. The process of discrete vibrational mode excitation in ϕ⁴ model is considered. The role of these oscillations in the creation of kink and antikink is discussed. Numerical results are presented as well as some attempts of analytical explanations. An intriguing fractal structure in parameter space dividing regions with and without creation is also presented.