Volume 49, Issue 11, November 2008
Index of content:
 ANNOUNCEMENTS


Announcement: Special issue on “Integrable Quantum Systems and Solvable Statistical Mechanics Models”
View Description Hide Description

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


On states of perfect correlation
View Description Hide DescriptionFor finitedimensional systems, states satisfying perfect correlation are just maximally entangled states that play an essential role in quantum informationtheory. Thus perfect correlation may be a key to entanglement phenomena. For general systems, Einstein–Podolsky–Rosen type states are characterized by perfect correlation and their properties are discussed. Conditions equivalent to perfect correlation are found. As an application of perfect correlation, all Einstein–Podolsky–Rosen type states on a twoparticle system are given. Unlike in finitedimensional systems, they are generally not quasiequivalent.

Monogamy equality in quantum systems
View Description Hide DescriptionThere is an interesting property about multipartite entanglement, called the monogamy of entanglement. The property can be shown by the monogamy inequality, called the Coffman–Kundu–Wootters inequality [Phys. Rev. A61, 052306 (2000); Coffman–Kundu–WoottersPhys. Rev. Lett.96, 220503 (2006)], and more explicitly by the monogamy equality in terms of the concurrence and the concurrence of assistance, , in the threequbit system. In this paper, we consider the monogamy equality in quantum systems. We show that if and only if and also show that if , then , while there exists a state in a system such that but .

Semispectral measures as convolutions and their moment operators
View Description Hide DescriptionThe moment operators of a semispectral measure having the structure of the convolution of a positive measure and a semispectral measure are studied, paying attention to the natural domains of these unbounded operators. The results are then applied to conveniently determine the moment operators of the Cartesian margins of the phase space observables.

On the structure of Clifford quantum cellular automata
View Description Hide DescriptionWe study reversible quantum cellular automata with the restriction that these are also Clifford operations. This means that tensor products of Pauli operators (or discrete Weyl operators) are mapped to tensor products of Pauli operators. Therefore Clifford quantum cellular automata are induced by symplectic cellular automata in phase space. We characterize these symplectic cellular automata and find that all possible local rules must be, up to some global shift, reflection invariant with respect to the origin. In the onedimensional (1D) case we also find that every uniquely determined and translationally invariant stabilizer state can be prepared from a product state by a single Clifford cellular automaton time step, thereby characterizing this class of stabilizer states, and we show that all 1D Clifford quantum cellular automata are generated by a few elementary operations. We also show that the correspondence between translationally invariant stabilizer states and translationally invariant Clifford operations holds for periodic boundary conditions.

 GENERAL RELATIVITY AND GRAVITATION


Effective field equations of braneinduced electromagnetism
View Description Hide DescriptionUsing a covariant embedding formalism, we find the effective field equations for the electromagnetism that emerge on branes in the context of Dvali–Gabadadze–Porrati (DGP) braneworld scenario. Our treatment is essentially geometrical. We start with Maxwell equations in five dimensions and project them into an arbitrary brane. The formalism is quite general and allows us to consider curved bulk spaces and curved branes whose tension is not necessarily null. The kinetic electromagnetic term induced on the world volume of the brane, proper of DGP models, is incorporated in this formulation by means of an appropriate match condition. We also give an estimate of each term of the effective field equations and determine the domain in which the fourdimensional Maxwell equations can be recovered in the brane.

Anisotropic cosmological models with spinor and scalar fields and viscous fluid in presence of a term: Qualitative solutions
View Description Hide DescriptionThe study of a selfconsistent system of interacting spinor and scalar fields within the scope of a Bianchi type I (BI) gravitational field in the presence of a viscous fluid and term has been carried out. The system of equations defining the evolution of the volume scale of BI universe, energy density, and corresponding Hubble constant has been derived. The system in question has been thoroughly studied qualitatively. Corresponding solutions are graphically illustrated. The system in question is also studied from the view point of blow up. It has been shown that the blow up takes place only in the presence of viscosity.

 DYNAMICAL SYSTEMS


Melnikov theory to all orders and Puiseux series for subharmonic solutions
View Description Hide DescriptionWe study the problem of subharmonic bifurcations for analytic systems in the plane with perturbations depending periodically on time, in the case in which we only assume that the subharmonic Melnikov function has at least one zero. If the order of zero is odd, then there is always at least one subharmonic solution, whereas if the order is even, in general, other conditions have to be assumed to guarantee the existence of subharmonic solutions. Even when such solutions exist, in general, they are not analytic in the perturbation parameter. We show that they are analytic in a fractional power of the perturbation parameter. To obtain a fully constructive algorithm which allows us not only to prove existence but also to obtain bounds on the radius of analyticity and to approximate the solutions within any fixed accuracy, we need further assumptions. The method we use to construct the solution—when this is possible—is based on a combination of the Newton–Puiseux algorithm and the tree formalism. This leads to a graphical representation of the solution in terms of diagrams. Finally, if the subharmonic Melnikov function is identically zero, we show that it is possible to introduce higher order generalizations, for which the same kind of analysis can be carried out.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Superintegrable threebody systems on the line
View Description Hide DescriptionWe consider classical threebody interactions on a Euclidean line depending on the reciprocal distance of the particles and admitting four functionally independent quadratic in the momentum first integrals. These systems are multiseparable, superintegrable, and equivalent (up to rescalings) to a oneparticle system in the threedimensional Euclidean space. Common features of the dynamics are discussed. We show how to determine quantum symmetry operators associated with the first integrals considered here but do not analyze the corresponding quantum dynamics. The conformal multiseparability is discussed and examples of conformal first integrals are given. The systems considered here in generality include the Calogero, Wolfes, and other threebody interactions widely studied in mathematical physics.

 FLUIDS


Analytical solutions to the Navier–Stokes equations
View Description Hide DescriptionWith the previous results for the analytical blowup solutions of the dimensional Euler–Poisson equations, we extend the same structure to construct an analytical family of solutions for the isothermal Navier–Stokes equations and pressureless Navier–Stokes equations with densitydependent viscosity.

 STATISTICAL PHYSICS


Biorthonormal eigenbasis of a Markovian master equation for the quantum Brownian motion
View Description Hide DescriptionThe solution to a quantum Markovian master equation of a harmonic oscillator weakly coupled to a thermal reservoir is investigated as a nonHermitian eigenvalue problem in space coordinates. In terms of a pair of quantum actionangle variables, the equation becomes separable and a complete set of biorthogonal eigenfunctions can be constructed. Properties of quantum states, such as the change in the quantum coherence length, damping in the motion, and disappearance of the spatial interference pattern, can then be described as the decay of the nonequilibrium modes in the eigenbasis expansion. It is found that the process of gaining quantum coherence from the environment takes a longer time than the opposite process of losing quantum coherence to the environment. An estimate of the time scales of these processes is obtained.

A connected graph identity and convergence of cluster expansions
View Description Hide DescriptionRecently Fernández and Procacci found an estimate on the radius of convergence of the cluster expansion for a gas of particles on a discrete space. Their estimate is an improvement on previous results of Kotecký and Preiss and of Dobrushin. This note presents a connected graph identity and uses it to prove a version of their estimate that applies to considerably more general contexts, including that where the space is not discrete.

Transport and bistable kinetics of a Brownian particle in a nonequilibrium environment
View Description Hide DescriptionA system reservoir model, where the associated reservoir is modulated by an external colored random force, is proposed to study the transport of an overdamped Brownian particle in a periodic potential. We then derive the analytical expression for the average velocity, mobility, and diffusion rate. The bistable kinetics and escape rate from a metastable state in the overdamped region are studied consequently. By numerical simulation we then demonstrate that our analytical escape rate is in good agreement with that of the numerical result.

 METHODS OF MATHEMATICAL PHYSICS


On the spectral theory and dispersive estimates for a discrete Schrödinger equation in one dimension
View Description Hide DescriptionBased on the recent work [Komech et al., “Dispersive estimates for 1D discrete Schrödinger and KleinGordon equations,” Appl. Anal.85, 1487 (2006)] for compact potentials, we develop the spectral theory for the onedimensional discrete Schrödinger operator, . We show that under appropriate decay conditions on the general potential (and a nonresonance condition at the spectral edges), the spectrum of consists of finitely many eigenvalues of finite multiplicities and the essential (absolutely continuous) spectrum, while the resolvent satisfies the limiting absorption principle and the Puiseux expansions near the edges. These properties imply the dispersive estimates for any fixed and any , where denotes the spectral projection to the absolutely continuous spectrum of . In addition, based on the scattering theory for the discrete Jost solutions and the previous results by Stefanov and Kevrekidis [“Asymptotic behaviour of small solutions for the discrete nonlinear Schrödinger and KleinGordon equations,” Nonlinearity18, 1841 (2005)], we find new dispersive estimates , which are sharp for the discrete Schrödinger operators even for .

The higher flows of harmonic maps and an application to the Virasoro action
View Description Hide DescriptionOne characteristic of soliton equations is their infinite number of conservation laws. These conservation laws give rise to a hierarchy of partial differential equations. One example is the AKNS hierarchy. Terng and Uhlenbeck, Poisson Actions and Scattering Theory for Integrable Systems, Surveys in Differential Geometry: Integrable Systems (A supplement to J. Diff. Geom.), 4 (1998) described the conservation laws that give rise to the AKNS hierarchy. This hierarchy can also be realized in a loop group setting, where the equations are called the positive flows. The name arises from the power of which generates each flow. Terng (Loop Groups and Integrable Systems, Preliminary notes) also showed that in some sense, the harmonic map equation arises as a flow in this hierarchy. The harmonic map equation also has a hierarchy of negative flows. However, these are not differential equations like the positive flows. It remained to be asked how the positive and negative flows relate and if they commute. The answer to the latter question is in the positive. As for the first question, the two hierarchies are part of one larger hierarchy. This was solved by using a more complex loop group and explicitly determining the loop algebra splitting. This splitting has applications to physics and is applied to an example of Schwarz involving the Virasoro action in Sec. VIII.

Classification of irreducible weight modules over algebra
View Description Hide DescriptionIn this paper, we show that the support of an irreducible weight module over the algebra , which has an infinite dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite dimensional. As a corollary, we obtain that every irreducible weight module over the algebra , having a nontrivial finite dimensional weight space, is a Harish–Chandra module (and hence is either an irreducible highest or a lowest weight module or an irreducible module of the intermediate series).

On the blowup phenomena of the periodic Dullin–Gottwald–Holm equation
View Description Hide DescriptionWe first present several new blowup results for the periodic Dullin–Gottwald–Holm equation by using some new Sobolev inequalities. We then investigate the blowup rate for all nonglobal solutions and determine the blowup set of blowingup solutions to the equation for a large class of initial data. We finally address the lower semicontinuity of the existence time of solutions with smooth initial data.

The functional integral with unconditional Wiener measure for anharmonic oscillator
View Description Hide DescriptionIn this article we propose the calculation of the unconditional Wiener measure functional integral with a term of the fourth order in the exponent by an alternative method as in the conventional perturbative approach. In contrast to the conventional perturbation theory, we expand into power series the term linear in the integration variable in the exponent. In such a case we can profit from the representation of the integral in question by the parabolic cylinder functions. We show that in such a case the series expansions are uniformly convergent and we find recurrence relations for the Wiener functional integral in the dimensional approximation. In continuum limit we find that the generalized Gelfand–Yaglom differential equation with solution yields the desired functional integral (similarly as the standard Gelfand–Yaglom differential equation yields the functional integral for linear harmonic oscillator).

Absolutely continuous spectrum of a Schrödinger operator on a tree
View Description Hide DescriptionWe give sufficient conditions for the presence of the absolutely continuous spectrum of a Schrödinger operator on a regular rooted tree without loops (also called regular Bethe lattice or Cayley tree).

The Cauchy problem of the Ward equation with mixed scattering data
View Description Hide DescriptionWe solve the Cauchy problem of the Ward equation with both continuous and discrete scattering data.
