Volume 47, Issue 6, June 2006
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Time minimal trajectories for a spin particle in a magnetic field
View Description Hide DescriptionIn this paper we consider the minimum time population transfer problem for the component of the spin of a (spin 1/2) particle, driven by a magnetic field, that is constant along the axis and controlled along the axis, with bounded amplitude. On the Bloch sphere (i.e., after a suitable Hopf projection), this problem can be attacked with techniques of optimal syntheses on twodimensional (2D) manifolds. Let be the two energy levels, and the bound on the field amplitude. For each couple of values and , we determine the time optimal synthesis starting from the level , and we provide the explicit expression of the time optimal trajectories, steering the state one to the state two, in terms of a parameter that can be computed solving numerically a suitable equation. For , every time optimal trajectory is bangbang and, in particular, the corresponding control is periodic with frequency of the order of the resonance frequency . On the other side, for , the time optimal trajectory steering the state one to the state two is bangbang with exactly one switching. For fixed , we also prove that for the time needed to reach the state two tends to zero. In the case there are time optimal trajectories containing a singular arc. Finally, we compare these results with some known results of Khaneja, Brockett, and Glaser and with those obtained by controlling the magnetic field both on the and directions (or with one external field, but in the rotating wave approximation). As a byproduct we prove that the qualitative shape of the time optimal synthesis presents different patterns that cyclically alternate as , giving a partial proof of a conjecture formulated in a previous paper.

Absence of reflection as a function of the coupling constant
View Description Hide DescriptionWe consider solutions of the onedimensional equation where is locally integrable, is integrable with supp, and is a coupling constant. Given a family of solutions which satisfy for all , we prove that the zeros of , the Wronskian of and , form a discrete set unless . Setting , one sees that a particular consequence of this result may be stated as: if the fixed energy scattering experiment gives rise to a reflection coefficient which vanishes on a set of couplings with an accumulation point, then .

Complex periodic potentials with a finite number of band gaps
View Description Hide DescriptionWe obtain several new results for the complex generalized associated Lamé potential , where , is the Jacobi elliptic function with modulus parameter , and there are four real parameters . First, we derive two new duality relations which, when coupled with a previously obtained duality relation, permit us to relate the band edge eigenstates of the 24 potentials obtained by permutations of the parameters . Second, we pose and answer the question: how many independent potentials are there with a finite number “” of band gaps when are integers and ? For these potentials, we clarify the nature of the band edge eigenfunctions. We also obtain several analytic results when at least one of the four parameters is a halfinteger. As a byproduct, we also obtain new solutions of Heun’s differential equation.

A geometrical characterization of commutative positive operator valued measures
View Description Hide DescriptionWe show that a POV measure on the Borel algebra of the reals is commutative if and only if there exists a PV measure on and, for every in the spectrum of , a probability measure on such that the effect coincides with , where is the selfadjoint operator associated to . The relevance of this result to the theory of the sharp reconstruction is analyzed.

Spectral geometry of Minkowski space
View Description Hide DescriptionAfter recalling Snyder’s idea [Phys. Rev.71, 38 (1947)] of using vector fields over a smooth manifold as “coordinates on a noncommutative space,” we discuss a twodimensional toymodel whose “dual” noncommutative coordinates form a Lie algebra: this is the wellknown Minkowski space [Phys. Lett. B334, 348 (1994)]. We show how to improve Snyder’s idea using the tools of quantum groups and noncommutative geometry. We find a natural representation of the coordinate algebra of Minkowski as linear operators on an Hilbert space (a major problem in the construction of a physical theory), study its “spectral properties,” and discuss how to obtain a Dirac operator for this space. We describe two Dirac operators. The first is associated with a spectral triple. We prove that the cyclic integral of Dimitrijevic et al. [Eur. Phys. J. C31, 129 (2003)] can be obtained as Dixmier trace associated to this triple. The second Dirac operator is equivariant for the action of the quantum Euclidean group, but it has unbounded commutators with the algebra.

GHZ extraction yield for multipartite stabilizer states
View Description Hide DescriptionLet be an arbitrary stabilizer state distributed between three remote parties, such that each party holds several qubits. Let be a stabilizer group of . We show that can be converted by local unitaries into a collection of singlets, GHZ states, and local onequbit states. The numbers of singlets and GHZs are determined by dimensions of certain subgroups of . For an arbitrary number of parties we find a formula for the maximal number of partite GHZ states that can be extracted from by local unitaries. A connection with earlier introduced measures of multipartite correlations is made. An example of an undecomposable fourparty stabilizer state with more than one qubit per party is given. These results are derived from a general theoretical framework that allows one to study interconversion of multipartite stabilizer states by local Clifford group operators. As a simple application, we study threeparty entanglement in twodimensional lattice models that can be exactly solved by the stabilizer formalism.

Probability and Quantum Symmetries. II. The Theorem of Nœther in quantum mechanics
View Description Hide DescriptionFor the largest class of physical systems having a classical analog, a new rigorous, but not probabilistic, Lagrangian version of nonrelativistic quantum mechanics is given, in terms of a notion of regularized action function. As a consequence of the study of the symmetries of this action, an associated Nœther theorem is obtained. All the quantum symmetries resulting from the canonical quantization procedure follow in this way, as well as a number of symmetries which are new even for the case of the simplest systems. The method is based on the study of a corresponding Lie algebra and an analytical continuation in the time parameter of the probabilistic construction given in paper I of this work. Generically, the associated quantum first integrals are time dependent and the probabilistic model provides a natural interpretation of the new symmetries. Various examples illustrate the physical relevance of our results.

The weak Hopf algebras related to generalized KacMoody algebra
View Description Hide DescriptionWe define a kind of quantized enveloping algebra of a generalized KacMoody algebra by adding a generator satisfying for some integer . We denote this algebra by . This algebra is a weak Hopf algebra if and only if . In general, it is a bialgebra, and contains a Hopf subalgebra. This Hopf subalgebra is isomorphic to the usually quantum envelope algebra of a generalized KacMoody algebra.

Analytic calculation of energies and wave functions of the quartic and pure quartic oscillators
View Description Hide DescriptionGround state energies and wave functions of quartic and pure quartic oscillators are calculated by first casting the Schrödinger equation into a nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). In the QLM the nonlinear differential equation is solved by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. Our explicit analytic results are then compared with exact numerical and also with WKB solutions and it is found that our ground statewave functions, using a range of small to large coupling constants, yield a precision of between 0.1 and 1 percent and are more accurate than WKB solutions by two to three orders of magnitude. In addition, our QLM wave functions are devoid of unphysical turning point singularities and thus allow one to make analytical estimates of how variation of the oscillator parameters affects physical systems that can be described by the quartic and pure quartic oscillators.

The Bargmann transform and regularization of the 2, 3, 5dimensional Kepler problem
View Description Hide DescriptionWe introduce a Bargmann transform for the space of square integrable functions on the dimensional unit sphere inmersed in . This is done on base of the Hopf fibration for the spheres with and a suitable canonical transformation relating two different ways to regularize the dimensional Kepler problem (with fixed negative energy) involving the null complex quadric inmersed in . We prove the unitarity of the Bargmann transform onto a suitable space of analytical functions. We give reproducing kernels for these spaces of analytical functions which, for the cases , are defined as the kernel of quantizations of restrictions for regularizations of the classical Kepler problem. We give an inversion formula for our Bargmann transform. We also give a set of coherent states for , and their semiclassical asymptotics . Our Bargmann transform is actually a coherent statestransform. Additionally, we use the moment map technique in order to construct a map with values in that gives the base to define our canonical transformation.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Asymptotic iteration method solutions to the relativistic DuffinKemmerPetiau equation
View Description Hide DescriptionA simple exact analytical solution of the relativistic DuffinKemmerPetiau equation within the framework of the asymptotic iteration method is presented. Exact bound state energy eigenvalues and corresponding eigenfunctions are determined for the relativistic harmonic oscillator as well as the Coulomb potentials. As a nontrivial example, the anharmonic oscillator is solved and the energy eigenvalues are obtained within the perturbation theory using the asymptotic iteration method.

More anomaly free models of sixdimensional gauged supergravity
View Description Hide DescriptionWe construct a huge number of anomaly free models of sixdimensional gauged supergravity. The gauge groups are products of and , and every hyperino is charged under some of the gauge groups. It is also found that the potential may have flat directions when the symmetry is diagonally gauged together with another gauge group. In an Appendix, we determine the contribution to the global anomaly from symplectic Majorana Weyl fermions in six dimensions.

Character formulas and partition functions in higher dimensional conformal field theory
View Description Hide DescriptionA discussion of character formulas for positive energy, unitary irreducible representations of the conformal group is given, employing Verma modules and Weyl group reflections. Product formulas for various conformal group representations are found. These include generalizations of those found by Flato and Fronsdal for . In even dimensions the products for free representations split into two types depending on whether the dimension is divisible by four or not.
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 GENERAL RELATIVITY AND GRAVITATION


Optical geometry analysis of the electromagnetic selfforce
View Description Hide DescriptionWe present an analysis of the behavior of the electromagnetic selfforce for charged particles in a conformally static spacetime, interpreting the results with the help of optical geometry. Some conditions for the vanishing of the local terms in the selfforce are derived and discussed.

Parametrization of solutions of the Lewis metric by a Painlevé transcendent III
View Description Hide DescriptionWithout making use of the Ernst formalism we look directly for particular solutions of field equations describing stationary axisymmetric vacuum space–time using Weyl coordinates. The solutions that we obtain, by simple separation of variables, are parametrized in the general case by a III transcendent of Painlevé with two arbitrary constants.

On the geometry of Killing and conformal tensors
View Description Hide DescriptionThe second order Killing and conformal tensors are analyzed in terms of their spectral decomposition, and some properties of the eigenvalues and the eigenspaces are shown. When the tensor is of type I with only two different eigenvalues, the condition to be a Killing or a conformal tensor is characterized in terms of its underlying almostproduct structure. A canonical expression for the metrics admitting these kinds of symmetries is also presented. The space–time cases and are analyzed in more detail. Starting from this approach to Killing and conformal tensors a geometric interpretation of some results on quadratic first integrals of the geodesic equation in vacuum PetrovBel type D solutions is offered. A generalization of these results to a wider family of type D space–times is also obtained.

Walker’s theorem without coordinates
View Description Hide DescriptionWe provide a coordinatefree version of the local classification, due to Walker [Q. J. Math.1, 69 (1950)], of null parallel distributions on pseudoRiemannian manifolds. The underlying manifold is realized, locally, as the total space of a fiber bundle, each fiber of which is an affine principal bundle over a pseudoRiemannian manifold. All structures just named are naturally determined by the distribution and the metric, in contrast with the noncanonical choice of coordinates in the usual formulation of Walker’s theorem.

Twocomponent cosmological fluids with gravitational instabilities
View Description Hide DescriptionA survey of linearized cosmological fluid equations with a number of different matter components is made. To begin with, the onecomponent case is reconsidered to illustrate some important mathematical and physical points rarely discussed in the literature. The work of some previous studies of twocomponent systems are examined and reanalyzed to point out some deficiencies of solutions, and further solutions and physical interpretation are then presented. This leads into a general twocomponent model with variable velocity dispersion parameters and mass density fractions of each component. The equations, applicable to both hot dark matter(HDM) and cold dark matter(CDM) universes are solved in the long wavelength limit. This region is of interest, because some modes in this range of wave numbers are Jeans unstable. The mixture Jeans wave number of the twocomponent system is introduced and interpreted, and the solutions are discussed, particularly in comparison to analogous solutions previously derived for plasma modes. This work is applicable to that region in the early Universe , where large scale structure formation is thought to have occurred.

Short wavelength analysis of the evolution of perturbations in a twocomponent cosmological fluid
View Description Hide DescriptionThe equations describing a twocomponent cosmological fluid with linearized density perturbations are investigated in the small wavelength or large limit. The equations are formulated to include a baryonic component, as well as either a hot dark matter(HDM) or cold dark matter(CDM) component. Previous work done on such a system in static space–time is extended to reveal some interesting physical properties, such as the Jeans wave number of the mixture, and resonant mode amplitudes. A WKB technique is then developed to study the expanding universe equations in detail, and to see whether such physical properties are also of relevance in this more realistic scenario. The Jeans wave number of the mixture is reinterpreted for the case of an expanding background space–time. The various modes are obtained to leading order, and the amplitudes of the modes are examined in detail to compare to the resonances observed in the static space–time results. It is found that some conclusions made in the literature about static space–time results cannot be carried over to an expanding cosmology.
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 DYNAMICAL SYSTEMS


Eigenvalues of zero energy in the linearized NLS problem
View Description Hide DescriptionWe study a pair of neutrally stable eigenvalues of zero energy in the linearized NLS equation. We prove that the pair of isolated eigenvalues, where each eigenvalue has geometric multiplicity one and algebraic multiplicity , is associated with negative eigenvalues of the energy operator, where if is even and or if is odd. When the potential of the linearized NLS problem is perturbed due to parameter continuations, we compute the exact number of unstable eigenvalues that bifurcate from the neutrally stable eigenvalues of zero energy.
