Table of contents

We discuss the first passage time problem in the semi-infinite interval, for homogeneous stochastic Markov processes with Lévy stable jump length distributions λ(x) ∼ ℓ^α/|x|^1+α (|x| ≫ ℓ), namely, Lévy flights (LFs). In particular, we demonstrate that the method of images leads to a result, which violates a theorem due to Sparre Andersen, according to which an arbitrary continuous and symmetric jump length distribution produces a first passage time density (FPTD) governed by the universal long-time decay ∼t^−3/2. Conversely, we show that for LFs the direct definition known from Gaussian processes in fact defines the probability density of first arrival, which for LFs differs from the FPTD. Our findings are corroborated by numerical results.

We consider a network model, embedded on the Manhattan lattice, of a quantum localization problem belonging to symmetry class C. This arises in the context of quasiparticle dynamics in disordered spin-singlet superconductors which are invariant under spin rotations but not under time reversal. A mapping exists between problems belonging to this symmetry class and certain classical random walks which are self-avoiding and have attractive interactions; we exploit this equivalence, using a study of the classical random walks to gain information about the corresponding quantum problem. In a field-theoretic approach, we show that the interactions may flow to one of two possible strong-coupling regimes separated by a transition: however, using Monte Carlo simulations we show that the walks are in fact always compact two-dimensional objects with a well-defined one-dimensional surface, indicating that the corresponding quantum system is localized.

A spherical Hopfield-type neural network is introduced, involving neurons and patterns that are continuous variables. Both the thermodynamics and dynamics of this model are studied. In order to have a retrieval phase a quartic term is added to the Hamiltonian. The thermodynamics of the model is exactly solvable and the results are replica symmetric. A Langevin dynamics leads to a closed set of equations for the order parameters and effective correlation and response function typical for neural networks. The stationary limit corresponds to the thermodynamic results. Numerical calculations illustrate these findings.

A recently introduced extension of the corner transfer matrix renormalization group method useful for the study of self-avoiding walk-type models is presented in detail and applied to a class of interacting self-avoiding walks due to Blöte and Nienhuis. This model displays two different types of collapse transition depending on model parameters. One is the standard θ-point transition. The other is found to give rise to a first-order collapse transition despite being known to be in other respects critical.

We show how the Lyapunov exponents of a dynamic system can, in general, be expressed in terms of the free energy of a (non-Hermitian) quantum many-body problem. This puts their study as a problem of statistical mechanics, whose intuitive concepts and techniques of approximation can hence be borrowed.

We define pseudo-reality and pseudo-adjointness of a Hamiltonian, H, as ρHρ⁻¹ = H* and μHμ⁻¹ = H', respectively. We prove that the former yields the necessary condition for a spectrum to be real whereas the latter helps in fixing a definition for the inner-product of the eigenstates. Here we separate out the adjointness of an operator from its Hermitian adjointness. It turns out that a Hamiltonian possessing a real spectrum is first pseudo-real, further it could be Hermitian, PT-symmetric or pseudo-Hermitian.

Diagonalization of a certain operator in irreducible representations of the positive discrete series of the quantum algebra U_q(su_1,1) is studied. Spectrum and eigenfunctions of this operator are explicitly found. These eigenfunctions, when normalized, constitute an orthonormal basis in the representation space. The initial U_q(su_1,1) basis and the basis of these eigenfunctions are interconnected by a matrix with entries expressed in terms of big q-Laguerre polynomials. The unitarity of this connection matrix leads to an orthogonal system of functions, which are dual with respect to big q-Laguerre polynomials. This system of functions consists of two separate sets of functions, which can be expressed in terms of q-Meixner polynomials M_n(x; b, c; q) either with positive or negative values of the parameter b. The orthogonality property of these two sets of functions follows directly from the unitarity of the connection matrix. As a consequence, one obtains an orthogonality relation for the q-Meixner polynomials M_n(x; b, c; q) with b < 0. A biorthogonal system of functions (with respect to the scalar product in the representation space) is also derived.

The symmetry groups of double-wall carbon nanotubes (DWCNs) are the line and point groups for the commensurate and incommensurate walls, respectively. For the tubes with diameters between 2.8 Å and 50 Å all possible DWCNs are found. Among them all the 318 commensurate DWCNs are singled out and their symmetry groups are calculated. DWCNs are low symmetry objects with respect to the constituent single-wall tubes, and this symmetry reduction is described by the symmetry breaking groups. Both symmetry and symmetry breaking groups affect the physical properties of the DWCNs. While, e.g., quantum numbers and selection rules are related to the symmetry group, the low interaction between the walls is determined by the breaking group.

The Schwinger representation and the Marumori–Yamamura–Tokunaga boson expansion are used to describe the Lipkin model in terms of generalized coherent states. The ground-state energy is obtained within several variant types of coherent states. It has been found that two of the kinds considered describe particularly well the transition from weak to strong coupling, providing a remarkable improvement of the mean-field description of the transition zone. The time evolution predicted by generalized coherent states has been investigated numerically by comparing its dynamics with the exact one.

In the context of a two-parameter (α, β) deformation of the canonical commutation relation leading to nonzero minimal uncertainties in both position and momentum, the harmonic oscillator spectrum and eigenvectors are determined by using an extension of the techniques of conventional supersymmetric quantum mechanics (SUSYQM) combined with shape invariance under parameter scaling. The resulting supersymmetric partner Hamiltonians correspond to different masses and frequencies. The exponential spectrum is proved to reduce to a previously found quadratic spectrum whenever one of the parameters α, β vanishes, in which case shape invariance under parameter translation occurs. In the special case where α = β ≠ 0, the oscillator Hamiltonian is shown to coincide with that of the q-deformed oscillator with q > 1 and its eigenvectors are therefore n-q-boson states. In the general case where 0 ≠ α ≠ β ≠ 0, the eigenvectors are constructed as linear combinations of n-q-boson states by resorting to a Bargmann representation of the latter and to q-differential calculus. They are finally expressed in terms of a q-exponential and little q-Jacobi polynomials.

In this paper certain 'fermionic' Stirling numbers introduced recently are discussed. Roughly speaking, these numbers are obtained by taking the 'fermionic' limit q → −1 of the q-deformed Stirling numbers. The usual Stirling numbers correspond in this language to the 'bosonic' limit q → 1. It is shown that the fermionic Stirling numbers are given by binomial coefficients and that they satisfy the same relations as the undeformed Stirling numbers. The fermionic relatives of Lah numbers are also very briefly discussed.

We apply the newly derived number-operational phase entangled state to solve a number-phase entangled Jaynes–Cummings model. The time evolution of the phase and number difference is calculated. The model also exhibits some collapse-revival phenomena.

Inspired by factorized scattering from δ-type impurities in (1 + 1)-dimensional spacetime, we propose and analyse a generalization of the Zamolodchikov–Faddeev algebra. Distinguished elements of the new algebra, called reflection and transmission generators, encode the particle–impurity interactions. We describe in detail the underlying algebraic structure. The relative Fock representations are explicitly constructed and a general factorized scattering theory is developed in this framework.