Table of contents

We consider a bistable piecewise potential subject to both a periodic signal and a random force. The output signal-to-noise ratio (SNR) is calculated for a small-amplitude periodic signal of arbitrary frequency and noise strength D. Our analysis not only recovers the well known non-monotonic dependence of SNR on D at fixed , but also predicts the non-monotonic dependence of SNR on at fixed D. The latter phenomenon does not appear in the commonly assumed adiabatic approximation which applies only at low frequencies.

Stochastic systems with a stretched exponential form of probability density function (turbulence etc) are studied. A dimensionless moments is defined as , where is the standard moment of order p (p > n). Pseudo-scaling (PS) is defined as the existence of power relationships , where exponent depends on p,q and n. The scaling is purely due to the way the given quantities are constructed and to the existence of the stretched exponential decay of the probability density functions. It is shown that for large enough n,p,q the pseudo-scaling takes place even if the ordinary scaling is broken. It is shown that there are two kinds of asymptotic pseudo-scaling: with and with . If ordinary scaling also takes place in the system, then these two kinds of pseudo-scaling lead to two kinds of corresponding ordinary scaling laws. Agreement between the theoretical approach and experimental results of different authors is established for turbulent systems (laboratory and numerical simulations), both for situations where ordinary scaling takes place and for situations where ordinary scaling does not take place.

The solid-on-solid model provides a commonly used framework for the description of surfaces. In recent it has been extended in order to investigate the effect of defects in the bulk on the roughness of the surface. The determination of the ground state of this model leads to a combinatorial problem, which is reduced to an uncapacitated, convex minimum-circulation problem. We will show that the successive shortest path algorithm solves the problem in polynomial time.

We study a one-dimensional driven lattice gas model in which quenched random jump rates are associated with the particles. Under suitable conditions on the distribution of jump rates the model displays a phase transition from a high-density `laminar' phase with product measure to a low-density `jammed' phase in which the interparticle spacings have no stationary distribution. Using a waiting time representation the phase transition is shown to be equivalent to a pinning transition of directed polymers with columnar defects. The phenomenon is argued to have a natural realization in traffic flow.

Binary sequences with low autocorrelations are important in communication engineering and in statistical mechanics as ground states of the Bernasconi model. Computer searches are the main tool in the construction of such sequences. Owing to the exponential size of the configuration space, exhaustive searches are limited to short sequences. We discuss an exhaustive search algorithm with run-time characteristic and apply it to compile a table of exact ground states of the Bernasconi model up to N = 48. The data suggest F > 9 for the optimal merit factor in the limit .

The full spectrum of transfer matrices of the general eight-vertex model on a square lattice is obtained by numerical diagonalization. The eigenvalue spacing distribution and the spectral rigidity are analysed. In non-integrable regimes we have found eigenvalue repulsion as for the Gaussian orthogonal ensemble in random matrix theory. By contrast, in integrable regimes we have found eigenvalue independence leading to a Poissonian behaviour, and, for some points, level clustering. These first examples from classical statistical mechanics suggest that the conjecture of integrability successfully applied to quantum spin systems also holds for classical systems.

The Lax pair formulation is presented for completely integrable quantum lattice spin open chains. Specifically, the Lax pair for the one-dimensional Heisenberg XXZ open chain is explicitly constructed. Our construction provides an alternative and direct demonstration of the quantum integrability of the system.

We have recently used unitarity and cross-unitarity properties of an symmetric R-matrix to construct the transfer matrix t(u) for an N-site open spin chain. Here we give a fusion procedure for such a chain, and we prove that the fused transfer matrix is commutative with the original transfer matrix t(u).

We consider a microwave resonator with three single-channel waveguides attached. One of these serves to couple waves into and out of the resonator; the remaining two are connected to form a one-way handle so as to break time reversal invariance. The poles of the input - output scattering coefficient of such a resonator are shown to be the eigenvalues of a non-Hermitian effective `Hamiltonian' , the anti-Hermitian part of which has rank 1 and is responsible for the breaking of time reversal invariance. All of the spectral statistics recently observed for such a microwave billiard are reproduced quantitatively by taking H and as random matrices. In particular, the distribution of nearest-neighbour spacings of the resonances is close to that of the GUE when H belongs to the GOE corresponding to a Sinai shape of the resonator; linear level repulsion results when H belongs to the Poissonian ensemble as it corresponds to a rectangular resonator.

We study the qualitative change of the dynamics of a generalized two-dimensional quadratic map under the influence of parametric perturbations which operate in the chaotic parameter set. It is shown that such perturbations can lead to the suppression of chaos and appearance of a regular (periodic) behaviour. Numerically we can argue that the suppression of chaos due to the parametric excitation is caused by a shift of the windows of periodic behaviour in the bifurcation diagram.

We present a detailed study of the nearest-neighbour ferromagnetic Ising model on a Cayley tree. In the limit of zero field, the system displays glassy behaviour below a crossover temperature, , that scales inversely with the logarithm of the number of generations; thus is inversely proportional to the logarithm of the logarithm of the number of sites. Non-Gaussian magnetization distributions are observed for , reminiscent of that associated with the central spin of the Edwards - Anderson model on the same tree; furthermore, a dynamical study indicates metastability, long relaxation times and ageing consistent with the development of glassy behaviour for a finite but macroscopic number of sites.

We introduce a class of random spin systems without frustration, which satisfies the conditions for gauge symmetry. The method of gauge transformation provides several properties on the phase diagram of gauge symmetric models; this method is almost rigorous, while the derivation for the absence of re-entrance contains unproved assumption. In the present random models, the absence of re-entrance can be shown exactly. Furthermore, the phase diagram and the critical properties are exactly related with those in the original pure system. The present random systems correspond to the Mattis model with asymmetric bond distribution in the Ising case, and a kind of the gauge glass model in the XY case. In the case of the clock model in two dimensions, we find a new thermodynamic phase which has long-range spin-glass correlation with critical (power-decaying) ferromagnetic correlation.

We study the fluctuations that are predicted in the autocorrelation function of an energy eigenstate of a chaotic, two-dimensional billiard by the conjecture (due to Berry) that the eigenfunction is a Gaussian random variable. We find an explicit formula for the root-mean-square amplitude of the expected fluctuations in the autocorrelation function. These fluctuations turn out to be in the small (high energy) limit. For comparison, any corrections due to scars from isolated periodic orbits would also be . The fluctuations take on a particularly simple form if the autocorrelation function is averaged over the direction of the separation vector. We compare our various predictions with recent numerical computations of Li and Robnik for the Robnik billiard, and find good agreement. We indicate how our results generalize to higher dimensions.

The relativistic quantum statistical properties of a charged many-fermion system with single fermion rest mass confined in a mesoscopic ring are investigated. The small ring is threaded by an Aharonov - Bohm magnetic flux . We have performed analytical calculations starting from evaluating the logarithm of the grand partition functions by using the Mellin transform. The persistent currents, number densities, total energies, heat capacity and hence the fluctuations of the total energies are obtained both in the weakly degenerate and degenerate situations for finite temperature. These thermodynamic functions oscillate periodically in with period , and they are related to the modified Bessel functions of the second kind . For the weakly degenerate case, where the chemical potential , we find they decay exponentially with respect to the decrease in temperature, and their magnitudes also decrease in the form of modified exponential functions as the circumference L increases. In the degenerate case, where , these functions behave complicatedly in the form of generalized hypergeometric functions, which decay in the power series forms at very low temperature. The non-relativistic limits are given by considering the asymptotic behaviours of .

Using the boundary Yang - Baxter equations and exact results on the bulk S-matrices, we compute exact boundary scattering amplitudes of the supersymmetric sine - Gordon model with integrable boundary potentials.

The non-classical feature of sub-Poissonian photon statistics is extended from one- to two-mode electromagnetic fields, incorporating the physically motivated property of invariance under passive unitary transformations. Applications to squeezed coherent states, squeezed thermal states, and superposition of coherent states are given. Dependences of extent of non-classical behaviour on the independent squeezing parameters are graphically displayed.

We consider two analytic representations of the SU(1,1) Lie group: the representation in the unit disc based on the SU(1,1) Perelomov coherent states and the Barut - Girardello representation based on the eigenstates of the SU(1,1) lowering generator. We show that these representations are related through a Laplace transform. A `weak' resolution of the identity in terms of the Perelomov SU(1,1) coherent states is presented which is valid even when the Bargmann index k is smaller than . Various applications of these results in the context of the two-photon realization of SU(1,1) in quantum optics are also discussed.

The inner - outer part factorization of analytic representations in the unit disc is used for an effective characterization of the number-phase statistical properties of a quantum harmonic oscillator. It is shown that the factorization is intimately connected with the number-phase Weyl semigroup and its properties. In the Barut - Girardello analytic representation the factorization is implemented as a convolution. Several examples are given which demonstrate the physical significance of the factorization and its role for quantum statistics. In particular, we study the effect of phase-space interference on the factorization properties of a superposition state.

The following two claims have been put forward in support of an individual interpretation of quantum mechanics: within the framework of quantum measurement theory, (i) the concept of measurement of a sharp observable can be completely characterized in terms of the notion of calibration, and (ii) the frequency interpretation of probability (of a measurement outcome) can be founded on the notion of a property of an individual system. Here we set out to generalize these results to measurement schemes which may be unitary or non-unitary and to object and pointer observables which may be sharp or unsharp.

We construct a q-deformation of the centrally-extended Schrödinger algebra and an algebraic representation theory through lowest-weight representations. We use Verma modules over , calculate their singular vectors and factorize the Verma modules by submodules built on the singular vectors. We also give a realization of with q-difference operators and obtain a polynomial realization of the lowest-weight representations and an infinite family of q-difference equations which may be called generalized q-deformed heat/Schrödinger equations. We also apply our methods to the on-shell q-Schrödinger algebra proposed by Floreanini and Vinet.

Potential symmetries, which are not local symmetries, are carried out for the porous medium equation where when it can be written in a conserved form. These symmetries are realized as local symmetries of a related auxiliary system, and lead to the construction of corresponding invariant solutions, as well as to the linearization of the equation by non-invertible mappings.

Motivated by the aim of finding generalized transformation coefficients for the symmetric group, we calculate the matrix which transforms the basis functions of the Young - Yamanouchi basis into the basis functions of its dual. Our approach is to derive the representation matrices for both bases and then determine the transformation matrix. The dual basis is associated with the subgroup chain , whereas the usual YY basis is associated with the subgroup chain . A combinatorial technique, jeu de taquin, is used to define the basis, via the Young - Yamanouchi symbols and Young tableaux with which the basis functions can be indexed.

We use B-type knot theory to find new solutions of Sklyanin's reflection equation in a systematic way. This generalizes the well known Baxterization of Birman - Wenzl algebras and should describe integrable systems which are restricted to a half plane.

The Wigner - Inonu contraction from the rotation group O(3) to the Euclidean group E(2) is used to relate the separation of variables in the Laplace - Beltrami operators on two corresponding homogeneous spaces. Different realizations of the contraction take the two separable coordinate systems on the sphere to the four on the plane .

The purpose of this article is the description of the classical dynamics of a resonant non-integrable Hamiltonian, which is written in the form

and where the term with behaves as a perturbation to the remaining integrable part (call it ) of the Hamiltonian. Apart from the chaotic region around the separatrix of , the dynamics of H is clearly different from that of only in the neighbourhood of low-order periodic orbits of , where coupling-induced resonance islands are seen to emerge. In order to model these resonance islands, is Taylor expanded in terms of its action integrals (thanks to recent exact analytical calculations) and the perturbation with is Fourier expanded in terms of the angles conjugate to the actions of . Retaining in the expansion only the term which is almost secular (because of the vicinity of the periodic orbit) leads to a local single resonance form of H. The classical frequencies and action integrals, which can be calculated analytically for this local expression of H, are shown to be in excellent agreement with `exact' numerical values deduced from power spectra and Poincaré surfaces of section. It is pointed out in the discussion that all the trajectories inside coupling-induced resonance islands share one almost degenerate classical frequency, and that the width of the coupling-induced island grows as the square root of the perturbation parameter , but is inversely proportional to the square root of the slow classical frequency at the periodic orbit and to the square root of the derivative, with respect to the first action integral, of the winding number.

We consider a set of localized Wannier functions which can be used to describe the Bloch bands of one-dimensional Hamiltonians whose associated Weyl function is periodic under a hexagonal lattice of translations in phase space. These Hamiltonians can be used to describe electrons on two-dimensional hexagonal lattices penetrated by a magnetic field. The effect on the Wannier functions of a rotation of the phase plane is considered. The resulting transformation is found to be surprisingly complicated when the quantized Hall conductance integer for the Bloch band is non-zero.

Dromion structures have been found in many (2 + 1)-dimensional models. The similar structure in (3 + 1) dimensions is studied in this paper for a KdV-type equation, . Starting from the bilinear form of the model, we found that there are five types of multi-dromion solutions for its potentials, say, . The first type of multi-dromion solution is driven by multi-camber solitons with one of them being non-parallel to the x-axis. The second, third and fourth types of multi-dromion solutions are driven by two camber solitons and one, two and three sets of parallel plane solitons, respectively. The fifth type of multi-dromion solution is driven by one set of parallel plane solitons, one set of camber solitons which are parallel to the x-axis and one camber soliton which is non-parallel to the x-axis. Phase shifts may be involved in the interactions among the multi-dromions for the first and fifth types of solutions. A single dromion solution of the model may possess quite a free shape. For instance, the point-like dromions, ring-type dromions, extended and sharp dromions and oscillatory dromions can be obtained by selecting some arbitrary functions appropriately.

Two new exact solutions for the bouncer model are found and their relations to some solvable and non-solvable Lie algebras are shown.

Prüfer's spectral algorithm is applied to the classical analogue of Schrödinger's equation. The discrete spectrum is interpreted as a bifurcation phenomenon caused by two simultaneous classical motions: rotation and squeezing. The energy eigenvalues coincide with the bifurcation parameters for the classical orbits.

Equations for relativistic particles for arbitrary spin have been of interest since Dirac original work for spin , but they involved either bothersome constraints or start with as many Dirac equations as are required to get the derived spin from its original value. We first show that it is possible to have just one equation involving 's and 's matrices that give possibilities up to for the spin. We then decompose the 's and 's into direct products of ordinary spin matrices and a new type of them that we call sign spin. The problem reduces then to one in terms of the generators of a U(4) group entirely similar to the one in the spin - isospin theory of nuclear physics and hence the name of supermultiplets in the title. Using then the techniques of the latter we discuss the problem of a free particle in a magnetic field for n = 1,2 and 3 or equivalently eigenvalues for spins 0,, 1 and , and the energies are given as solutions of elementary algebraic equations.

A new method for deriving universal matrices from braid-group representation is discussed. In this case, universal operators can be defined and expressed in terms of products of braid-group generators. The advantage of this method is that matrix elements of are rank independent, and leaves multiplicity problem-concerning coproducts of the corresponding quantum groups untouched. As examples, -matrix elements of , , , and with multiplicity two for -type and for -type, -type, and -type quantum groups, which are related to Hecke algebra and Birman - Wenzl algebra, respectively, are derived by using this method.

Using our calculus for SU(3) a generating function for Wigner D-matrix elements is derived for the first time. From this an explicit expression for the individual matrix elements, in an arbitrary irreducible representation, is obtained, also for the first time, in terms of polynomials in the matrix elements of the defining representation of SU(3). This expression does not depend on any particular parametrization of the group.

The general four-parameter point interaction in one-dimensional quantum mechanics is regulated. It allows the exact solution, but not the perturbative one. We conjecture that this is due to the interaction not being asymptotically free. We then propose a different breakup of unperturbed theory and interaction, which now is asymptotically free but leads to the same physics. The corresponding regulated potential can be solved both exactly and perturbatively, in agreement with the conjecture.

The semiclassical quantization of sawtooth maps is reconsidered from the viewpoint of the higher-order terms in the asymptotic series. We carry out the semiclassical quantization in the leading and next leading orders. However, we fail in both cases. The reasons are discussed in detail.

We investigate Wilczeck's mutual fractional statistical model at the field-theoretical level. The effective Hamiltonian for the particles is derived by the canonical procedure, whereas the commutators of the anyonic excitations are proved to obey the Zamolodchikov - Faddeev algebra. Cases leading to well known statistics as well as Laughlin's wavefunction are discussed.

The Boyer - Finley equation, or Toda equation is both a reduction of the self-dual Einstein equations and the dispersionless limit of the 2D Toda lattice equation. This suggests that there should be a dispersive version of the self-dual Einstein equation which both contains the Toda lattice equation and whose dispersionless limit is the familiar self-dual Einstein equation. Such a system is studied in this paper. The results are achieved by using a deformation, based on an associative -product, of the algebra used in the study of the undeformed, or dispersionless, equations.

An R-matrix theory for the Dirac equation is shown to exist in spite of incompleteness of a relativistic R-matrix basis on the reaction surface, a phenomenon which does not occur in the non-relativistic case. The theory is constructed for the most general boundary conditions imposed on expansion basis functions. It is shown that the incompleteness of the expansion basis on the reaction surface results in a matrix correction appearing in the eigenfunction expansion of the R-matrix. The correction vanishes in the non-relativistic limit. The approach is applied to the relativistic generalizations of the Kapur - Peierls and Wigner resonance reaction theories.

A unitary transformation is introduced to determine the exact solution of the time-dependent quantum system. As examples, two quantum systems with Hamiltonian including the quadratic term of generators of SU(1,1) are solved analytically.