Volume 43, Issue 7, July 2002
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Timeenergy coherent states and adiabatic scattering
View Description Hide DescriptionCoherent states in the timeenergy plane provide a natural basis to study adiabatic scattering. We relate the (diagonal) matrix elements of the scattering matrix in this basis with the frozen onshell scattering data. We describe an exactly solvable model, and show that the error in the frozen data cannot be estimated by the Wigner time delay alone. We introduce the notion of energy shift, a conjugate of Wigner time delay, and show that for incoming state the energy shift determines the outgoing state.

Coherent state realizations of on the ntorus
View Description Hide DescriptionWe obtain a new family of coherent state representations of in which the coherent states are Wigner functions over a subgroup of For representations of of the type (λ, 0, 0,…), the basis functions are simple products of n exponential. The corresponding coherent state representations of the algebra are also obtained, and provide a polar decomposition of for any The modules thus obtained are useful in understanding contractions of and phase states of quantum optics.

Integrable variant of the onedimensional Hubbard model
View Description Hide DescriptionA new integrable model which is a variant of the onedimensional Hubbard model is proposed. The integrability of the model is verified by presenting the associated quantum matrix which satisfies the Yang–Baxter equation. We argue that the new model possesses the SO(4) algebra symmetry, which contains a representation of the ηpairing SU(2) algebra and a spin SU(2) algebra. Additionally, the algebraic Bethe ansatz is studied by means of the quantum inverse scattering method. The spectrum of the Hamiltonian, eigenvectors, as well as the Bethe ansatz equations, are discussed.

Renormalization analysis of correlation properties in a quasiperiodically forced twolevel system
View Description Hide DescriptionWe give a rigorous renormalization analysis of the selfsimilarity of correlation functions in a quasiperiodically forced twolevel system. More precisely, the system considered is a quantum twolevel system in a timedependent field consisting of periodic kicks with amplitude given by a discontinuous modulation function driven in a quasiperiodic manner at golden mean frequency. Mathematically, our analysis consists of a description of all piecewiseconstant periodic orbits of an additive functional recurrence. We further establish a criterion for such orbits to be globally bounded functions. In a particular example, previously only treated numerically, we further calculate explicitly the asymptotic height of the main peaks in the correlation function.

Nfold supersymmetry in quantum mechanics—analyses of particular models
View Description Hide DescriptionWe investigate particular models which can be Nfold supersymmetric at specific values of a parameter in the Hamiltonians. The models to be investigated are a periodic potential and a paritysymmetric sextic triplewell potential. Through the quantitative analyses on the nonperturbative contributions to the spectra by the use of the valley method, we show how the characteristic features of Nfold supersymmetry which have been previously reported by the authors can be observed. We also clarify the difference between quasiexactly solvable and quasiperturbatively solvable cases in view of the dynamical property, that is, dynamical Nfold supersymmetry breaking.

Quantum jump dynamics in cavity QED
View Description Hide DescriptionWe study the stochastic dynamics of the electromagnetic field in a lossless cavityinteracting with a beam of twolevel atoms, given that the atomic states are measured after they have crossed the cavity. The atoms first interact at the exit of the cavity with a classical laser field E and then enter into a detector which measures their states. Each measurement disentangles the field and the atoms and changes in a random way the state of the cavity field. For weak atomfield coupling, the evolution of when many atoms cross the cavity and the detector is characterized by a succession of quantum jumps occurring at random times, separated by quasiHamiltonian evolutions, both of which depend on the laser field E. For E=0, the dynamics is the same as in the Monte Carlo wave function model of Dalibard et al. [Phys. Rev. Lett. 68, 580 (1992)] and Carmichael, An Open System Approach to Quantum Optics, Lecture Notes in Physics Vol. 18 (Springer, Berlin, 1991)]. The density matrix of the quantum field, obtained by averaging the projector over all results of the measurements, is independent of E and follows the master equation of the damped harmonic oscillator at finite temperature. We provide numerical evidence showing that for large E, an arbitrary initial field state ψ(0)〉 evolves under the monitoring of the atoms and the measurements toward squeezed states moving in the αcomplex plane but with almost constant squeezing parameters and φ. The values of and φ are determined analytically. On the other hand, for E=0, the dynamics transforms the initial state into Fock states with fluctuating numbers of photons as shown in Kist et al. [J. Opt. B: Quantum Semiclassical Opt. 1, 251 (1999)]. In the last part, we derive the quantum jump dynamics from the linear quantum jump model proposed in Spehner and Bellissard [J. Stat. Phys. 104, 525 (2001)], for arbitrary open quantum systems having a Lindbladtype evolution. A careful derivation of the infinite jump rates limit, where the dynamics can be approximated by a diffusion process of the quantum state, is also presented.

Quantum models related to fouled Hamiltonians of the harmonic oscillator
View Description Hide DescriptionWe study a pair of canonoid (fouled) Hamiltonians of the harmonic oscillator which provide, at the classical level, the same equation of motion as the conventional Hamiltonian. These Hamiltonians, say and result to be explicitly time dependent and can be expressed as a formal rotation of two cubic polynomial functions, and of the canonical variables (q,p). We investigate the role of these fouled Hamiltonians at the quantum level. Adopting a canonical quantization procedure, we construct some quantum models and analyze the related eigenvalue equations. One of these models is described by a Hamiltonian admitting infinite selfadjoint extensions, each of them has a discrete spectrum on the real line. A selfadjoint extension is fixed by choosing the spectral parameter ε of the associated eigenvalue equation equal to zero. The spectral problem is discussed in the context of three different representations. For the eigenvalue equation is exactly solved in all these representations, in which squareintegrable solutions are explicitly found. A set of constants of motion corresponding to these quantum models is also obtained. Furthermore, the algebraic structure underlying the quantum models is explored. This turns out to be a nonlinear (quadratic) algebra, which could be applied for the determination of approximate solutions to the eigenvalue equations.
 Top

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Redefining spinors in Lorentzviolating quantum electrodynamics
View Description Hide DescriptionAn analysis of spinor redefinitions in the context of the Lorentzviolating quantum electrodynamics(QED) extension is performed. Certain parameters that apparently violate Lorentz invariance are found to be physically irrelevant as they can be removed from the Lagrangian using an appropriate redefinition of the spinor field components. It is shown that conserved currents may be defined using a modified action of the complex extension of the Lorentz group on the redefined spinors. This implies a natural correspondence between the apparently Lorentzviolating theory and conventional QED. Redefinitions involving derivatives are shown to relate certain terms in the QED extension to Lagrangians involving nonlocal interactions or skewed coordinate systems. The redundant parameters in the QED extension are identified and the Lagrangian is rewritten in terms of physically relevant coupling constants. The resulting Lagrangian contains only physically relevant parameters and transforms conventionally under Lorentz transformations.

Phase space structure and short distance behavior of local quantum field theories
View Description Hide DescriptionIn this article a general relation between the short distance structure of quantum field theories and their phase space properties is exemplified by the simple class of generalized free fields. As is known, theories with decent phase space properties, resulting from a finite or moderately increasing infinite particle spectrum, always have nontrivial scaling (short distance) limits [cf. D. Buchholz and R. Verch, Rev. Math. Phys. 10, 775 (1998)]. But, whereas in the finite particle case the phase space properties of the limit theories comply with strong nuclearity conditions, they violate in the infinite particle case even rather mild compactness assumptions. These results provide further evidence to the effect that relevant information on the short distance structure of a theory can be obtained by phase space analysis.
 Top

 GENERAL RELATIVITY AND GRAVITATION


Conditions for the alignment of the principal null directions of two Weyllike tensors
View Description Hide DescriptionA possible means to classify the interaction between the Weyl and Ricci tensors is to look at the number of principal null directions that the Weyl and Plebanski tensors have in common. This paper presents algebraic conditions that can be used to determine this number without explicitly calculating the principal null directions themselves.
 Top

 DYNAMICAL SYSTEMS


Modifying fractal basin boundaries by reshaping periodic terms
View Description Hide DescriptionA generic route is described for the modification of fractal basin boundaries in nonlinear systems by changing only the shape of a periodic (autonomous or nonautonomous) term in the dynamics equations. Two examples are used to illustrate the route: a noninvertible twodimensional map, and a driven dissipative oscillator with a cubic potential that typically models a metastable system close to a fold.

Complete sets of invariants for dynamical systems that admit a separation of variables
View Description Hide DescriptionConsider a classical Hamiltonian in dimensions consisting of a kinetic energy term plus a potential. If the associated Hamilton–Jacobi equation admits an orthogonal separation of variables, then it is possible to generate algorithmically a canonical basis Q, P where are the other secondorder constants of the motion associated with the separable coordinates, and The functions form a basis for the invariants. We show how to determine for exactly which spaces and potentials the invariant is a polynomial in the original momenta. We shed light on the general question of exactly when the Hamiltonian admits a constant of the motion that is polynomial in the momenta. For we go further and consider all cases where the Hamilton–Jacobi equation admits a secondorder constant of the motion, not necessarily associated with orthogonal separable coordinates, or even separable coordinates at all. In each of these cases we construct an additional constant of the motion.

Conservation laws for a class of nonlinear equations with variable coefficients on discrete and noncommutative spaces
View Description Hide DescriptionThe conservation laws for a class of nonlinear equations with variable coefficients on discrete and noncommutative spaces are derived. For discrete models the conserved charges are constructed explicitly. The applications of the general method include equations on quantum plane, supersymmetric equations for chiral and antichiral supermultiplets, auxiliary equations of integrable models which means various cases of nonlinear Toda lattice equations and anomalous diffusionequation.

Boundary values as Hamiltonian variables. II. Graded structures
View Description Hide DescriptionIt is shown that the new formula for the field theory Poisson brackets arises naturally in the proposed extension of the formal variational calculus incorporating divergences. The linear spaces of local functionals, evolutionary vector fields, functional forms, multivectors and differential operators become graded with respect to divergences. The bilinear operations, such as the action of vector fields onto functionals, the commutator of vector fields, the interior product of forms and vectors and the Schouten–Nijenhuis bracket are compatible with the grading. A definition of the adjoint graded operator is proposed and antisymmetric operators are constructed with the help of boundary terms. The fulfilment of the Jacobi identity for the new Poisson brackets is shown to be equivalent to vanishing of the Schouten–Nijenhuis bracket of the Poisson bivector with itself.

Boundary values as Hamiltonian variables. III. Ideal fluid with a free surface
View Description Hide DescriptionAn application of the approach to Hamiltonian treatment of boundary terms proposed in previous articles of this series is considered. Here the Hamiltonian formalism is constructed and the role of standard boundary conditions is revealed for a inviscid compressible fluid with surface tension which moves in a field of the Newtonian gravitational potential. It is shown that these boundary conditions guarantee absence of singular contributions to the equations of motion, i.e., to the Hamiltonian vector field. From the other side the Hamiltonian variation contains a nonzero boundary term. Such Hamiltonians are usually treated as “nondifferentiable” or “inadmissible.” We conclude that nondifferentiable functionals can be admissible Hamiltonians for nonultralocal Poisson brackets. We give a foursided picture of free surfacedynamics: both in Lagrangian and in Eulerian variables and also both in variational and in Hamiltonian approaches.
 Top

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


On Keller theorem for anisotropic media
View Description Hide DescriptionThe Keller theorem in the problem of effective conductivity in anisotropic twodimensional (2D) manycomponent composites makes it possible to establish a simple inequality for the isotropic part of the second rank symmetric tensor of effective conductivity.

Resonance dynamics and partial averaging in a restricted threebody system
View Description Hide DescriptionBased on the value of the orbital eccentricity of an object and also its proximity to the exact resonant orbit in a threebody system, the pendulum approximation [S. F. Dermott and C. D. Murray, Nature (London) 319, 201 (1983)] or the second fundamental model of resonance [M. H. Andoyer, Bull. Astron. 20, 321 (1903); J. Henrard and A. Lemaı̂tre, Celest. Mech. 30, 197 (1983)] are commonly used to study the motion of that object near its resonant state. In this paper, we present the method of partial averaging as an analytical approach to study the dynamical evolution of a body near a resonance. To focus attention on the capabilities of this technique, a restricted, circular and planar threebody system is considered and the dynamics of its outer planet while captured in a resonance with the inner body is studied. It is shown that the firstorder partially averaged system resembles a mathematical pendulum whose librational motion can be viewed as a geometrical interpretation of the resonance capture phenomenon. The driving force of this pendulum corresponds to the gravitational attraction of the inner body and its contribution, at different resonant states, is shown to be proportional to where is the order of the resonance and is the orbital eccentricity of the outer planet. As examples of such systems, the cases of (1:1), (1:2), and (1:3) resonances are discussed and the results are compared with known planetary systems such as the Sun–Jupiter–Trojan asteroids.
 Top

 STATISTICAL PHYSICS


A mathematical problem of the theory of gelation
View Description Hide DescriptionCurrent theory of gelation describes this process in terms of a set of nonlinear integral equations. In this article the uniqueness of nontrivial solutions of these equations within the unit functional hypercube has been proved. Besides, the convergence to this solution of iterations from an arbitrary point of the above hypercube has been established, which is of utmost importance for calculations of particular gelation processes.

Quadratic replica coupling in the Sherrington–Kirkpatrick mean field spin glass model
View Description Hide DescriptionWe develop a very simple method to study the high temperature, or equivalently high external field, behavior of the Sherrington–Kirkpatrick mean fieldspin glass model. The basic idea is to couple two different replicas with a quadratic term, trying to push out the two replica overlap from its replica symmetric value. In the case of zero external field, our results reproduce the well known validity of the annealed approximation, up to the known critical value for the temperature. In the case of nontrivial external field, we can prove the validity of the Sherrington–Kirkpatrick replica symmetric solution up to a line, which falls short of the Almeida–Thouless line, associated to the onset of the spontaneous replica symmetry breaking, in the Parisi ansatz. The main difference with the method, recently developed by Michel Talagrand, is that we employ a quadratic coupling, and not a linear one. The resulting flow equations, with respect to the parameters of the model, turn out to be very simple, and the parameter region, where the method works, can be easily found in explicit terms. As a straightforward application of cavity methods, we show also how to determine free energy and overlap fluctuations, in the region where replica symmetry has been shown to hold. It is a major open problem to give a rigorous mathematical treatment of the transition to replica symmetry breaking, necessarily present in the model.
 Top

 METHODS OF MATHEMATICAL PHYSICS


Wave focusing on the line
View Description Hide DescriptionFocusing of waves in one dimension is analyzed for the plasmawave equation and the wave equation with variable speed. The existence of focusing causal solutions to these equations is established, and such wavesolutions are constructed explicitly by deriving an orthogonality relation for the timeindependent Schrödinger equation. The connection between wave focusing and inverse scattering is studied. The potential at any point is recovered from the incident wave that leads to focusing to that point. It is shown that focusing waves satisfy certain temporalantisymmetry and support properties. Discontinuities in the spatial and temporal derivatives of the focusing waves are examined and related to the discontinuities in the potential of the Schrödinger equation. The theory is illustrated with some explicit examples.
