Table of contents

The quantum states of a spinless charged particle on a hyperbolic plane in the presence of a uniform magnetic field with a generalized quantization condition are proved to be the bases of the irreducible Hilbert representation spaces of the Lie algebra u(1, 1). The dynamical symmetry group U(1, 1) with the explicit form of the Lie algebra generators is extracted. It is also shown that the energy has an infinite-fold degeneracy in each of the representation spaces which are allocated to the different values of the magnetic field strength. Based on the simultaneous shift of two parameters, it is also noted that the quantum states realize the representations of Lie algebra u(2) by shifting the magnetic field strength.

A deterministic and time reversible Bohmian mechanics for operators with continuous and discrete spectra is presented. Randomness enters only through initial conditions. Operators with discrete spectra are incorporated into Bohmian mechanics by associating with each operator a continuous variable in which a finite range of the continuous variable corresponds to the same discrete eigenvalue. In this way a deterministic and time reversible Bohmian mechanics can handle the creation and annihilation of particles. The formalism does not depend on the details of the Hamiltonian. Furthermore, many consistent choices are available for the dynamics. Examples are given and generalizations are discussed.

Ultra-discrete versions of the discrete Painlevé equations are well known. However, evidence for their integrability has so far been restricted. In this letter, we show that their Lax pairs can be constructed and, furthermore, that compatibility conditions of the result yield the ultra-discrete Painlevé equation. For conciseness, we restrict our attention to a new d-P_III.

A simple iterative method is described for finding the eigenvalues of a general square complex matrix. Several numerical examples involving complex symmetric matrices are treated. In particular, it is found that a naive matrix calculation without complex rotation produces resonant state energies in accord with those given by the recently introduced naive complex hypervirial perturbation theory.

We consider nematic liquid crystals in a bounded, convex polyhedron described by a director field n(r) subject to tangent boundary conditions. We derive lower bounds for the one-constant elastic energy in terms of topological invariants. For a right rectangular prism and a large class of topologies, we derive upper bounds by introducing test configurations constructed from local conformal solutions of the Euler–Lagrange equation. The ratio of the upper and lower bounds depends only on the aspect ratios of the prism. As the aspect ratios are varied, the minimum-energy conformal state undergoes a sharp transition from being smooth to having singularities on the edges.

A concrete proposal for the realization of a Bose–Einstein condensate of kicked rotators is presented. Studying their dynamics via the one-dimensional Gross–Pitaevskii equation on a ring we point out the existence of a Lax pair and an infinite countable set of conserved quantities. Under equal conditions we make numerical comparisons of the dynamics and their effective irreversibility in time, of ensembles of chaotic classical, and BECs of interaction-free quantum, and interacting quantum kicked rotators.

An amplitude-phase formula for the S-matrix due to a central potential is derived. The derivation makes use of invariants of the Ermakov–Lewis type.

We develop perturbation theory and physically motivated resummations of the perturbation theory for the problem of a tracer particle diffusing in a random medium. The random medium contains point scatterers of density ρ uniformly distributed throughout the material. The tracer is a Langevin particle subjected to the quenched random force generated by the scatterers. Via our perturbative analysis, we determine when the random potential can be approximated by a Gaussian random potential. We also develop a self-similar renormalization group approach based on thinning out the scatterers; this scheme is similar to that used with success for diffusion in Gaussian random potentials and agrees with known exact results. To assess the accuracy of this approximation scheme, its predictions are confronted with results obtained by numerical simulation.

The long-time dynamics of the 1D contact process suddenly brought out of an uncorrelated initial state is studied through a light-cone transfer-matrix renormalization group approach. At criticality, the system undergoes ageing which is characterized through the dynamical scaling of the two-times autocorrelation and autoresponse functions. The observed non-equality of the ageing exponents a and b excludes the possibility of a finite fluctuation–dissipation ratio in the ageing regime. The scaling form of the critical autoresponse function is in agreement with the prediction of local scale invariance.

The long-time dynamics of the critical contact process which is brought suddenly out of an uncorrelated initial state undergoes ageing in close analogy with quenched magnetic systems. In particular, we show through Monte Carlo simulations in one and two dimensions and through mean-field theory that time-translation invariance is broken and that dynamical scaling holds. We find that the autocorrelation and autoresponse exponents λ_Γ and λ_R are equal but, in contrast to systems relaxing to equilibrium, the ageing exponents a and b are distinct. A recent proposal to define a non-equilibrium temperature through the short-time limit of the fluctuation–dissipation ratio is therefore not applicable.

The generalized entropic measure, which is maximized by a given arbitrary distribution under the constraints on normalization of the distribution and the finite ordinary expectation value of a physical random quantity, is considered. To examine if it can be of physical relevance, its experimental robustness is discussed. In particular, Lesche's criterion is analysed, which states that an entropic measure is stable if its change under an arbitrary weak deformation of the distribution (representing fluctuations of experimental data) remains small. It is essential to note the difference between this criterion and thermodynamic stability. A general condition, under which the generalized entropy becomes stable, is derived. Examples known in the literature, including the entropy for the stretched-exponential distribution, the quantum-group entropy and the κ-entropy are discussed.

An explicit algorithm to provide the pruning front for the area-preserving Hénon map is presented. The procedure terminates within finitely many steps when the map has hyperbolic structure. The only information required to specify the pruning front is a bifurcation diagram of homoclinic orbits, and it is obtained by tracking orbits from the anti-integrable limit. The pruned region thus determined is used to construct the Markov partition of the map, and the topological entropy is evaluated as an application.

We calculate the grammatical complexity of the symbol sequences generated from the Hénon map and the Lozi map using the recently developed methods to construct the pruning front. When the map is hyperbolic, the language of symbol sequences is regular in the sense of the Chomsky hierarchy and the corresponding grammatical complexity takes finite values. It is found that the complexity exhibits a self-similar structure as a function of the system parameter, and the similarity of the pruning fronts is discussed as an origin of such self-similarity. For non-hyperbolic cases, it is observed that the complexity monotonically increases as we increase the resolution of the pruning front.

The completeness of the Barut–Girardello coherent states of the quantized su_q(1, 1) algebra is studied. Using the inverse Mellin transform the integration measure for the resolution of the unit operator is obtained. The measure is found to be positive-definite.

Geometrical properties of three-body orbits with zero angular momentum are investigated. If the moment of inertia is also constant along the orbit, the triangle whose vertexes are the positions of the bodies, and the triangle whose perimeters are the momenta of the bodies, are always similar ('synchronized similar triangles'). This similarity yields kinematic equalities between mutual distances and magnitude of momenta. Moreover, if the orbit is a solution to the equation of motion under a homogeneous potential, the orbit has a new constant involving momenta. For orbits with zero angular momentum and non-constant moment of inertia, we introduce the scaled variables, positions divided by square root of the moment of inertia and momenta derived from the velocity of the scaled positions. Then the similarity and the kinematic equalities hold for the scaled variables. Using this similarity, we prove that any bounded three-body orbit with zero angular momentum under a homogeneous potential whose degree is smaller than 2 has infinitely many collinear configurations (syzygies or eclipses) or collisions.

The traditional Tamm–Dancoff (TD) method is one of the standard procedures for solving the Schrödinger equation of fermion many-body systems. However, it meets a serious difficulty when an instability occurs in the symmetry-adapted ground state of the independent particle approximation (IPA) and when the stable IPA ground state becomes of broken symmetry. If one uses the stable but broken symmetry IPA ground state as the starting approximation, TD wave functions also become of broken symmetry. On the contrary, if we start from a symmetry-adapted but unstable wave function, the convergence of the TD expansion becomes bad. Thus, the requirements of symmetry and rapid convergence are not in general compatible in the conventional TD expansion of the systems with strong collective correlations. Along the same line as Fukutome's, we give a group-theoretical deduction of a U(n) dyadic TD equation by using a matrix-valued generator coordinate.

A unit-vector field n on a convex three-dimensional polyhedron is tangent if, on the faces of , n is tangent to the faces. A homotopy classification of tangent unit-vector fields continuous away from the vertices of is given. The classification is determined by certain invariants, namely edge orientations (values of n on the edges of ), kink numbers (relative winding numbers of n between edges on the faces of ), and wrapping numbers (relative degrees of n on surfaces separating the vertices of ), which are subject to certain sum rules. Another invariant, the trapped area, is expressed in terms of these. One motivation for this study comes from liquid crystal physics; tangent unit-vector fields describe the orientation of liquid crystals in certain polyhedral cells.

The two-soliton solution of the KdV equation is known not to degenerate into a solution describing resonant triads of solitons in 1 + 1 dimensions. However, we show that two integrable coupled-KdV systems of the Drinfeld–Sokolov class possess solutions which may degenerate into resonant triads. These solutions are associated with a resonance relation which generalizes the usual one previously considered by Hirota and Ito.

This work aims at the extension of the power series in R and ln R of interaction energy k and separation constant A for the system of two protons and one electron in the Born–Oppenheimer approximation. The wave equation is separated into confocal elliptic coordinates, in order to obtain two one-dimensional problems. The approach that we present here is based upon our previous calculations, where a relation between A and k for the (outer) ξ-equation was obtained in two different but substantially equivalent ways: either by an integral equation representation of the solution or by a logarithmic-perturbative expansion, in powers of the energy difference k, from the united atom state. At variance with our previous work, we make use of the (inner) η-equation of a remarkably simple determinantal equation proposed long ago by Hylleraas for which we develop a recursive method of calculation. By combining the two procedures we obtain the solution of the problem by solving for the coefficients of the R and ln R expansion. We calculate in this way coefficients up to O(R⁷) and show how higher order coefficients may be evaluated recursively.

Wu and Yu recently examined point interactions in one dimension in the form of the Fermi pseudo-potential. On the other hand there are point interactions in the form of self-adjoint extensions (SAEs) of the kinetic energy operator. We examine the relationship between the point interactions in these two forms in the one-channel and two-channel cases. In the one-channel case the pseudo-potential leads to the standard three-parameter family of SAEs. In the two-channel case the pseudo-potential furnishes a ten-parameter family of SAEs.

We study bipartite entanglement in a general one-particle state, and find that the linear entropy, quantifying the bipartite entanglement, is directly connected to the participation ratio, characterizing the state localization. The more extended the state, the more entangled it is. We apply the general formalism to investigate ground-state and dynamical properties of entanglement in the one-dimensional Harper model.

Following Shastry and Sutherland I construct the super Lax operators for the Calogero model in the oscillator potential. These operators can be used for the derivation of the eigenfunctions and integrals of motion of the Calogero model and its supersymmetric version. They allow us to infer several relations involving the Lax matrices for this model in a fast way. It is shown that the super Lax operators for the Calogero and Sutherland models can be expressed in terms of the supercharges and so-called local Dunkl operators constructed in our recent paper with M Ioffe. Several important relations involving Lax matrices and Hamiltonians of the Calogero and Sutherland models are easily derived from the properties of Dunkl operators.

We generalize the Brundobler–Elser hypothesis in the multistate Landau–Zener problem to the case when instead of a state with the highest slope of the diabatic energy level there is a band of states with an arbitrary number of parallel levels having the same slope. We argue that the probabilities of counterintuitive transitions among such states are exactly zero.

We arrive at a necessary and sufficient criterion that can be readily used for interconvertibility between general, all-tripartite Gaussian states under local quantum operation. The derivation involves a systematic reduction that converts the original complex conditions in high-dimensional, 6n × 6n matrix space eventually into 2 × 2 matrix problems.

We generalize classical Yang–Mills theory by extending nonlinear constitutive equations for Maxwell fields to non-Abelian gauge groups. Such theories may or may not be Lagrangian. We obtain conditions on the constitutive equations specifying the Lagrangian case, of which recently discussed non-Abelian Born–Infeld theories are particular examples. Some models in our class possess nontrivial Galilean (c → ∞) limits; we determine when such limits exist and obtain them explicitly.