Volume 45, Issue 11, November 2004
Index of content:
 METHODS OF MATHEMATICAL PHYSICS


Hopf maps as static solutions of the complex eikonal equation
View Description Hide DescriptionWe demonstrate that a class of torusshaped Hopf maps with arbitrary linking number obeys the static complex eikonal equation. Further, we explore the geometric structure behind these solutions, explaining thereby the reason for their existence. As this equation shows up as an integrability condition in certain nonlinear field theories, the existence of such solutions is of some interest.

 STATISTICAL PHYSICS


On adic model on the Cayley tree
View Description Hide DescriptionWe consider a nearestneighbor adic model with spin values on the Cayley tree of order . We prove that a adic Gibbs measure is unique for . If then we find a condition which guarantees uniqueness of adic Gibbs measure. Besides, the results are applied to the adic Ising model.

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


On extremal quantum states of composite systems with fixed marginals
View Description Hide DescriptionWe study the convex set of all bipartite quantum states with fixed marginal states and . The extremal states in this set have recently been characterized by Parthasarathy [Ann. Henri Poincaré (to appear), quantph/0307182]. Here we present an alternative necessary and sufficient condition for a state in to be extremal. Our approach is based on a canonical duality between bipartite states and a certain class of completely positive maps and has the advantage that it is easier to check and to construct explicit examples of extremal states. In dimension we give a simple new proof for the fact that all extremal states in are precisely the projectors onto maximally entangled wave functions. We also prove that in higher dimension this does not hold and construct an explicit example of an extremal state in that is not maximally entangled. Generalizations of this result to higher dimensions are also discussed.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


The Newtonian limit of the relativistic Boltzmann equation
View Description Hide DescriptionThe relativistic Boltzmann equation for a constant differential cross section and with periodic boundary conditions is considered. The speed of light appears as a parameter for a properly large and positive . A local existence and uniqueness theorem is proved in an interval of time independent of and conditions are given such that in the limit the solutions converge, in a suitable norm, to the solutions of the nonrelativistic Boltzmann equation for hard spheres.

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


Exact series solution to the two flavor neutrino oscillation problem in matter
View Description Hide DescriptionIn this paper, we present a real nonlinear differential equation for the two flavor neutrino oscillation problem in matter with an arbitrary density profile. We also present an exact series solution to this nonlinear differential equation. In addition, we investigate numerically the convergence of this solution for different matter density profiles such as constant and linear profiles as well as the Preliminary Reference Earth Model describing the Earth’s matter density profile. Finally, we discuss other methods used for solving the neutrino flavor evolution problem.

 STATISTICAL PHYSICS


The asymptotic behavior of the stochastic Ginzburg–Landau equation with multiplicative noise
View Description Hide DescriptionThe asymptotic behavior of the stochastic Ginzburg–Landau equation is studied. We obtain the stochastic Ginzburg–Landau equation as a finitedimensional random attractor.

 METHODS OF MATHEMATICAL PHYSICS


Umbral calculus, difference equations and the discrete Schrödinger equation
View Description Hide DescriptionIn this paper, we discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries. The method is then applied to the Schrödinger equation in order to obtain a realization of nonrelativistic quantum mechanics in discrete space–time. In this approach a quantum system on a lattice has a symmetry algebra isomorphic to that of the continuous case. Moreover, systems that are integrable, superintegrable or exactly solvable preserve these properties in the discrete case.

 STATISTICAL PHYSICS


Stability of spot and ring solutions of the diblock copolymer equation
View Description Hide DescriptionThe convergence theory shows that under certain conditions the diblock copolymer equation has spot and ring solutions. We determine the asymptotic properties of the critical eigenvalues of these solutions in order to understand their stability. In two dimensions a threshold exists for the stability of the spot solution. It is stable if the sample size is small and unstable if the sample size is large. The stability of the ring solutions is reduced to a family of finite dimensional eigenvalue problems. In one study no twointerface ring solutions are found by the convergence method if the sample is small. A stable twointerface ring solution exists if the sample size is increased. It becomes unstable if the sample size is increased further.

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


The Avez–Seifert theorem for the relativistic Lorentz force equation
View Description Hide DescriptionIn this paper we prove an extension of the Avez–Seifert theorem to the relativistic Lorentz force equation. Let be a globally hyperbolic space–time, an exact form on representing the electromagnetic field, the Lorentz force associated to , and a charge for a test particle. Let and be two chronologically related points on , then there exists a futurepointing timelike solution of the Lorentz force equation , connecting and .

 METHODS OF MATHEMATICAL PHYSICS


Covariants, joint invariants and the problem of equivalence in the invariant theory of Killing tensors defined in pseudoRiemannian spaces of constant curvature
View Description Hide DescriptionThe invariant theory of Killing tensors (ITKT) is extended by introducing the new concepts of covariants and joint invariants of (product) vector spaces of Killing tensors defined in pseudoRiemannian spaces of constant curvature. The covariants are employed to solve the problem of classification of the orthogonal coordinate webs generated by nontrivial Killing tensors of valence two defined in the Euclidean and Minkowski planes. Illustrative examples are provided.

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


On perturbations of Dirac operators with variable magnetic field of constant direction
View Description Hide DescriptionWe carry out the spectral analysis of matrix valued perturbations of threedimensional Dirac operators with variable magnetic field of constant direction. Under suitable assumptions on the magnetic field and on the pertubations, we obtain a limiting absorption principle, we prove the absence of singular continuous spectrum in certain intervals and state properties of the point spectrum. Various situations, for example, when the magnetic field is constant, periodic or diverging at infinity, are covered. The importance of an internaltype operator (a twodimensional Dirac operator) is also revealed in our study. The proofs rely on commutator methods.

Enhanced binding revisited for a spinless particle in nonrelativistic QED
View Description Hide DescriptionWe consider a spinless particle coupled to a quantized Bose field and show that such a system has a ground state for two classes of shortrange potentials which are alone too weak to have a zeroenergy resonance.

 METHODS OF MATHEMATICAL PHYSICS


Twodimensional Riemannian and Lorentzian geometries from secondorder ODE’s
View Description Hide DescriptionIn this paper we give an alternative geometrical derivation of the results recently presented by GarcíaGodínez, Newman, and SilvaOrtigoza on the class of all twodimensional Riemannian and Lorentzian metrics from secondorder ODE’s which are in duality with the twodimensional Hamilton–Jacobi equation. We show that, as it happens in the null surface formulation of general relativity, the Wünschmanntype condition can be obtained as a requirement of a vanishing torsion tensor. Furthermore, from these secondorder ODE's we obtain the associated Cartan connections.

 COMMENTS AND ERRATA



Addendum: Symmetries of the energymomentum tensor
View Description Hide DescriptionIn recent papers [J. Math. Phys. 44, 5142 (2003); 45, 1518 (2003); 45, 1532 (2004)] we have discussed matter symmetries of nonstatic spherically symmetric space–times, static plane symmetric space–times, and cylindrically symmetric static space–times. These have been classified for both cases when the energymomentum tensor is nondegenerate and also when it is degenerate. Here we add up some consequences and the missing references about the Ricci tensor.

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Almost sharp quantum effects
View Description Hide DescriptionQuantum effects are represented by operators on a Hilbert space satisfying and sharp quantum effects are represented by projection operators. We say that an effect is almost sharp if for projections and We give simple characterizations of almost sharp effects. We also characterize effects that can be written as longer products of projections. For generality we first work in the formalism of von Neumann algebras. We then specialize to the full operator algebra and to finite dimensional Hilbert spaces.

 METHODS OF MATHEMATICAL PHYSICS


Inhomogeneous quantum groups for particle algebras
View Description Hide DescriptionWe construct the inhomogeneous quantum groups and and investigate their Hopf algebrastructure. and leave the algebra of fermion and boson creation/annihilation operators invariant, respectively. We also present the corresponding matrices.

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


An algorithm for quaternionic linear equations in quaternionic quantum theory
View Description Hide DescriptionBy means of complex representation and companion vector, in this paper we introduce a definition of rank of a quaternion matrix, study the problems of quaternionic linear equations, and obtain an algorithm for quaternionic linear equations in quaternionic quantum theory.

 GENERAL RELATIVITY AND GRAVITATION


Intersecting hypersurfaces in dimensionally continued topological density gravitation
View Description Hide DescriptionWe consider intersecting hypersurfaces in curved spacetime with gravity governed by a class of actions which are topological invariants in lower dimensionality. Along with the Chern–Simons boundary terms there is a sequence of intersection terms that should be added in the action functional for a well defined variational principle. We construct them in the case of Characteristic Classes, obtaining relations which have a general topological meaning. Applying them on a manifold with a discontinuous connection 1form we obtain the gravity action functional of the system and show that the junction conditions can be found in a simple algebraic way. At the sequence of intersections there are localized independent energy tensors, constrained only by energy conservation. We work out explicitly the simplest nontrivial case.

 METHODS OF MATHEMATICAL PHYSICS


Asymptotic upper bounds for the entropy of orthogonal polynomials in the Szegő class
View Description Hide DescriptionWe give an asymptotic upper bound as for the entropy integral, where is the th degree orthonormal polynomial with respect to a weight on which belongs to the Szegő class. We also study two functionals closely related to the entropy integral. First, their asymptotic behavior is completely described for weights in the Bernstein class. Then, as for the entropy, we obtain asymptotic upper bounds for these two functionals when belongs to the Szegő class. In each case, we give conditions for these upper bounds to be attained.
