Table of contents

Following the quantization of a graph with the Dirac operator (spin-1/2), we explain how additional weights in the spectral form factor K(τ) due to spin propagation around orbits produce higher order terms in the small-τ asymptotics in agreement with symplectic random matrix ensembles. We determine conditions on the group of spin rotations sufficient to generate CSE statistics.

We present an analytical and rigorous study on a TOPS (task optimization in the presence of signal-dependent noise) model with a hold-on or an end-point control. Optimal control signals are rigorously obtained, which enables us to investigate various issues about the model including its trajectories, velocities, control signals, variances and the dependence of these quantities on various model parameters. With the hold-on control, we find that the optimal control can be implemented with an almost 'nil' hold-on period. The optimal control signal is a linear combination of two sub-control signals. One of the sub-control signals is positive and the other is negative. With the end-point control, the end-point variance is dramatically reduced, in comparison with the hold-on control. However, the velocity is not symmetric (bell shape). Finally, we point out that the velocity with a hold-on control takes the bell shape only within a limited parameter region.

In this paper, the subcritical, near-critical and supercritical asymptotic behaviour of a reversible Markov process as a chemical model for polymerization was studied. We establish the existence of three distinct stages (subcritical, near-critical and supercritical stages) of polymerization (in the thermodynamic limit as N → +∞), depending on the value of strength of the fragmentation reaction. These three stages correspond to the size of the largest length of polymers of size N to be itself of order log N, N^a(1/2 < a < 1) and N, respectively. Especially, a = 2/2σ+1 when the coagulation (reaction) rate constant R_ij of the polymers satisfies R_ij = i^σj^σ with 1/2 < σ < 3/2.

A coagulation–decoagulation model is introduced on a chain of length L with open boundary. The model consists of one species of particles which diffuse, coagulate and decoagulate preferentially in the leftward direction. They are also injected and extracted from the left boundary with different rates. We will show that on a specific plane in the space of parameters, the steady-state weights can be calculated exactly using a matrix product method. The model exhibits a first-order phase transition between a low-density and a high-density phase. The density profile of the particles in each phase is obtained both analytically and using the Monte Carlo simulation. The two-point density–density correlation function in each phase has also been calculated. By applying the Yang–Lee theory we can predict the same phase diagram for the model. This model is further evidence for the applicability of the Yang–Lee theory in the non-equilibrium statistical mechanics context.

I consider a physical system described by a continuous field theory and enclosed in a large but finite cubical box with periodic boundary conditions. The system is assumed to undergo a continuous phase transition at some critical point. The φ⁴ theory that is a continuous version of the Ising model is such a system but there are many other examples corresponding to higher spin, higher symmetry etc. The eigenfunctions of the corresponding Fokker–Planck (FP) operator can be chosen, of course, to be eigenfunctions of the momentum operator. It is shown that the eigenvalues of the FP operator, corresponding to each eigenvalue q of the momentum operator, evaluated at a transition point of the finite system, accumulate at zero, when the size of the system tends to infinity. There are many reasonable ways of defining a critical temperature of a finite system, which tends to the critical temperature of the infinite system as the size of the system tends to infinity. The accumulation of eigenvalues is neither affected by the specific choice of critical temperature of the finite system nor by whether the system is below or above its upper critical dimension.

We use first-order perturbation theory near the fermionic limit of the δ-function Bose gas in one dimension (i.e. a system of weakly interacting fermions) to study three situations of physical interest. The calculation is done using a pseudopotential which takes the form of a two-body δ''-function interaction. The three cases considered are the behaviour of the system with a hard wall, with a point where the strength of the pseudopotential changes discontinuously, and with a region of finite length where the pseudopotential strength is non-zero (this is sometimes used as a model for a quantum wire). In all cases, we obtain exact expressions for the density to first order in the pseudopotential strength. The asymptotic behaviour of the densities is in agreement with the results obtained from bosonization for a Tomonaga–Luttinger liquid, namely, an interaction dependent power-law decay of the density far from the hard wall, a reflection from the point of discontinuity and transmission resonances for the interacting region of finite length. Our results provide a non-trivial verification of the Tomonaga–Luttinger liquid description of the δ-function Bose gas near the fermionic limit.

We introduce a heterogeneous fractional Fokker–Planck equation (HFFPE) on heterogeneous fractal structure media describing systems involving external force fields. The HFFPE is shown to obey the generalized Einstein relation, and its stationary solution is the Boltzmann distribution. It is proven that the asymptotic shape of its solution is a stretched Gaussian and that its solution can be expressed in the form of a function of a dimensionless similarity variable for constant and generic potentials with polar singularity at origin.

We study harmonic polynomials on the quantum Euclidean space E^N_q generated by quantum coordinates x_i, i = 1, 2, ..., N, on which the quantum group SO_q(N) acts. They are defined as solutions of the equation Δ_qp = 0, where Δ_q is the q-Laplace operator on E^N_q. We construct a q-analogue of the classical zonal polynomials and associated spherical polynomials with respect to the quantum subgroup SO_q(N − 2). The associated spherical polynomials constitute an orthogonal basis of the spaces of homogeneous harmonic polynomials. They are represented as products of polynomials depending on q-radii and x_j, x_j', j' = N − j + 1. This representation is, in fact, a q-analogue of the classical separation of variables.

Nambu's construction of multi-linear brackets for super-integrable systems can be thought of as degenerate Poisson brackets with a maximal set of Casimirs in their kernel. By introducing privileged coordinates in phase space these degenerate Poisson brackets are brought to the form of Heisenberg's equations. We propose a definition for constructing quantum operators for classical functions, which enables us to turn the maximally degenerate Poisson brackets into operators. They pose a set of eigenvalue problems for a new state vector. The requirement of the single-valuedness of this eigenfunction leads to quantization. The example of the harmonic oscillator is used to illustrate this general procedure for quantizing a class of maximally super-integrable systems.

It is well known that the Schrödinger equation is equivalent to the wave equation for conservative bound quantum systems. Consequently, the motion of such a system is represented—from a mathematical point of view—by the motion of the characteristic surface of the wave equation. In this paper we present a demonstration of the periodic motion of the characteristic surface. It results that the normal curves of the characteristic surface are periodic solutions of the Hamilton–Jacobi equation written for the same system. This leads to a direct connection between the periodic solutions of the Hamilton–Jacobi equation and the wave properties of the system. The constants of motion corresponding to the above periodic solutions of the Hamilton–Jacobi equation are identical to the eigenvalues of the Schrödinger equation. These properties are proved without any approximation and they are valid for all the values of the principal quantum number.

A multi-shell generalization of a fermion representation of the q-deformed compact symplectic sp_q(4) algebra is introduced. An analytic form for the action of two or more generators of the Sp_q(4) symmetry on the basis states is determined and the result used to derive formulae for the overlap between number-preserving states as well as for matrix elements of a model Hamiltonian. A second-order operator in the generators of Sp_q(4) is identified that is diagonal in the basis set and that reduces to the Casimir invariant of the sp(4) algebra in the non-deformed limit of the theory. The results can be used in nuclear structure applications to calculate β-decay transition probabilities and to provide for a description of pairing and higher-order interactions in systems with nucleons occupying more than a single-j orbital.

The scattering properties of regularizing finite-range potentials constructed in the form of squeezed rectangles, which approximate the first and second derivatives of the Dirac delta function δ(x), are studied in the zero-range limit. Particularly, for a countable set of interaction strength values, a non-zero transmission through the point potential δ'(x), defined as the weak limit (in the standard sense of distributions) of a special dipole-like sequence of rectangles, is shown to exist when the rectangles are squeezed to zero width. A tripole sequence of rectangles, which gives in the weak limit the distribution δ''(x), is demonstrated to exhibit the total transmission for a countable sequence of the rectangle's width that tends to zero. However, this tripole sequence does not admit a well-defined point interaction in the zero-range limit, making sense only for a finite range of the regularizing rectangular-like potentials.

We show that the Fano operator for a quantum system confined to a line is uniquely determined by assuming reasonable behaviour under translation and parity transformation on phase space. In contrast, for a system on a lattice the same procedure does not work.

The symmetric version of the generalized Ginocchio potential, a member of the general exactly solvable Natanzon potential class, is analysed and its properties are compared with those of symmetric potentials from the more restricted shape-invariant class. It is found that the symmetric generalized Ginocchio potential has a number of properties in common with the latter potentials: it can be generated by an imaginary coordinate shift x → x + iε, its states are characterized by the quasi-parity quantum number, the spontaneous breakdown of symmetry occurs at the same time for all the energy levels and it has two supersymmetric partners which cease to be symmetric when the symmetry of the original potential is spontaneously broken.

We propose an efficient scheme for sharing a continuous-variable quantum secret using passive optical interferometry and squeezers: this efficiency is achieved by showing that a maximum of two squeezers is required to extract the secret state, and we obtain the cheapest configuration in terms of total squeezing cost. Squeezing is a cost for the dealer of the secret as well as for the receivers, and we quantify limitations to the fidelity of the extracted secret state in terms of the squeezing employed by the dealer.

Within the framework of -symmetric quantum mechanics of bound states (which works with parity-pseudo-Hermitian Hamiltonians H = H† and real spectra) we mimic some effects of the double-well structure of potentials by a pair of δ functions with mutually complex conjugate strengths. The model is solvable by the standard matching technique and exhibits several interesting features. We observe an amazingly robust absence of any -symmetry breaking. A quasi-degeneracy which occurs in the high-energy domain is interpreted as a manifestation of certain 'quantum beats'.

We study the conformal field theories corresponding to current superalgebras osp(2|2)⁽¹⁾_k and osp(2|2)⁽²⁾_k. We construct the free field realizations, screen currents and primary fields of these current superalgebras at general level k. All the results for osp(2|2)⁽²⁾_k are new, and the results for the primary fields of osp(2|2)⁽¹⁾_k also seem to be new. Our results are expected to be useful in the supersymmetric approach to Gaussian disordered systems such as the random bond Ising model and the Dirac model.

We reconsider the question of thermal radiation from accelerated mirrors in the model of Fulling and Davies, with the assumption that the mirror moves inertially after a finite acceleration time t_a. We obtain the Planck distribution by calculating the emitted spectrum as lim_ta→∞N_ω(t_a)/t_a, where N_ω(t_a) is the number of quanta produced in the ω mode on the trajectory with acceleration time t_a.