Volume 49, Issue 2, February 2008
Index of content:
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Timeofarrival probabilities and quantum measurements. II. Application to tunneling times
View Description Hide DescriptionWe formulate quantum tunneling as a timeofarrival problem: we determine the detection probability for particles passing through a barrier at a detector located a distance from the tunneling region. For this purpose, we use a positiveoperatorvalued measure (POVM) for the time of arrival determined in C. Anastopoulos and N. Savvidou, J. Math. Phys.47, 122106 (2006). This only depends on the initial state, the Hamiltonian, and the location of the detector. The POVM above provides a welldefined probability density and an unambiguous interpretation of all quantities involved. We demonstrate that for a class of localized initial states, the detection probability allows for an identification of tunneling time with the classic phase time. We also establish limits to the definability of tunneling time. We then generalize these results to a sequential measurement setup: the phasespace properties of the particles are determined by an unsharp sampling before their attempt to cross the barrier. For such measurements the tunneling time is defined as a genuine observable. This allows us to construct a probability distribution for its values that is definable for all initial states and potentials. We also identify a regime in which these probabilities correspond to a tunnelingtime operator.

Useful entanglement can be extracted from all nonseparable states
View Description Hide DescriptionWe consider entanglement distillation from a single copy of multipartite quantum states, and instead of rates we analyze the “quality” of the distilled entanglement. This quality is quantified by the fidelity with the gigahertz state. We show that each not fully separable state can increase the quality of the entanglement distilled from other states, no matter how weakly entangled is . We also generalize this to the case where the goal is distilling states different from the gigahertz. These results provide new insights on the geometry of the set of separable states and its dual (the set of entanglement witnesses).

Timeofarrival probabilities and quantum measurements. III. Decay of unstable states
View Description Hide DescriptionWe study the decay of unstable states by formulating quantum tunneling as a timeofarrival problem: we determine the detection probability for particles at a detector located a distance from the tunneling region. For this purpose, we use a positiveoperatorvalued measure (POVM) for the timeofarrival determined by Anastopoulos and Savvidou [J. Math. Phys.47, 122106 (2006)]. This only depends on the initial state, the Hamiltonian, and the location of the detector. The POVM above provides a welldefined probability density and an unambiguous interpretation of all quantities involved. We demonstrate that the exponential decay only arises if three specific mathematical conditions are met. Their physical content is the following: (i) the decay time is much larger than any microscopic timescale, so that the fine details of the initial state can be ignored, (ii) there is no quantum coherence between the different “attempts” of the particle to traverse the barrier, and (iii) the transmission probability varies little within the momentum spread of the initial state. We also determine the long time limits of the decay probability and we identify regimes, in which the decays have no exponential phase.

Gazeau–Klauder coherent states for trigonometric Rosen–Morse potential
View Description Hide DescriptionThe Gazeau–Klauder coherent states for the trigonometric Rosen–Morse potential are constructed. It is shown that the resolution of unity, temporal stability, and action identity conditions are satisfied for the coherent states. The Mandel parameter is also calculated for the weighting distribution function corresponding to the coherent states.

Inverse problems for the Schrödinger equations with timedependent electromagnetic potentials and the Aharonov–Bohm effect
View Description Hide DescriptionWe consider the inverse boundary value problem for the Schrödinger operator with timedependent electromagnetic potentials in domains with obstacles. We extend the resuls of the author’s works [Inverse Probl.19, 49 (2003);19, 985 (2003);20, 1497 (2004)] to the case of timedependent potentials. We relate our results to the Aharonov–Bohm effect caused by magnetic and electric fluxes.

OscillatorMorse–Coulomb mappings and algebras for constant or positiondependent mass
View Description Hide DescriptionThe boundstate solutions and the su(1,1) description of the dimensional radial harmonic oscillator, the Morse, and the dimensional radial Coulomb Schrödinger equations are reviewed in a unified way using the point canonical transformation method. It is established that the spectrumgenerating su(1,1) algebra for the first problem is converted into a potential algebra for the remaining two. This analysis is then extended to Schrödinger equations containing some positiondependent mass. The deformed su(1,1) construction recently achieved for a dimensional radial harmonic oscillator is easily extended to the Morse and Coulomb potentials. In the last two cases, the equivalence between the resulting deformed su(1,1) potential algebra approach and a previous deformed shape invariance one generalizes to a positiondependent mass background a wellknown relationship in the context of constant mass.

On the dimension of subspaces with bounded Schmidt rank
View Description Hide DescriptionWe consider the question of how large a subspace of a given bipartite quantum system can be when the subspace contains only highly entangled states. This is motivated in part by results of Hayden et al. [eprint arXiv:quantph∕0407049;Commun. Math. Phys., 265, 95 (2006)], which show that in large dimensional systems there exist random subspaces of dimension almost , all of whose states have entropy of entanglement at least . It is also a generalization of results on the dimension of completely entangled subspaces, which have connections with the construction of unextendible product bases. Here we take as entanglement measure the Schmidt rank, and determine, for every pair of local dimensions and , and every , the largest dimension of a subspace consisting only of entangled states of Schmidt rank or larger. This exact answer is a significant improvement on the best bounds that can be obtained using the random subspace techniques in Hayden et al. We also determine the converse: the largest dimension of a subspace with an upper bound on the Schmidt rank. Finally, we discuss the question of subspaces containing only states with Schmidt equal to .

Control landscapes for observable preparation with open quantum systems
View Description Hide DescriptionA quantum control landscape is defined as the observable as a function(al) of the system control variables. Such landscapes were introduced to provide a basis to understand the increasing number of successful experiments controlling quantum dynamics phenomena. This paper extends the concept to encompass the broader context of the environment having an influence. For the case that the open system dynamics are fully controllable, it is shown that the control landscape for open systems can be lifted to the analysis of an equivalent auxiliary landscape of a closed composite system that contains the environmental interactions. This inherent connection can be analyzed to provide relevant information about the topology of the original open system landscape. Application to the optimization of an observable expectation value reveals the same landscape simplicity observed in former studies on closed systems. In particular, no false suboptimal traps exist in the system control landscape when seeking to optimize an observable, even in the presence of complex environments. Moreover, a quantitative study of the control landscape of a system interacting with a thermal environment shows that the enhanced controllability attainable with open dynamics significantly broadens the range of the achievable observable values over the control landscape.

Direct demonstration of the completeness of the eigenstates of the Schrödinger equation with local and nonlocal potentials bearing a Coulomb tail
View Description Hide DescriptionDemonstrating the completeness of wave function solutions of the radial Schrödinger equation is a very difficult task. Existing proofs, relying on operator theory, are often very abstract and far from intuitive comprehension. However, it is possible to obtain rigorous proofs amenable to physical insight, if one restricts the considered class of Schrödinger potentials. One can mention, in particular, unbounded potentials yielding a purely discrete spectrum and shortrange potentials. However, those possessing a Coulomb tail, very important for physical applications, have remained problematic due to their longrange character. The method proposed in this paper allows to treat them correctly, provided that the nonCoulomb part of potentials vanishes after a finite radius. Nonlocality of potentials can also be handled. The main idea in the proposed demonstration is that regular solutions behave like sine/cosine functions for large momenta, so that their expansions verify Fourier transform properties. The highly singular point at of longrange potentials is dealt with properly using analytical properties of Coulomb wave functions. Lebesgue measure theory is avoided, rendering the demonstration clear from a physical point of view.

On spectra of Lüders operations
View Description Hide DescriptionWe show that all the eigenvalues of certain generalized Lüders operations are nonnegative real numbers in two cases of interest. In particular, given a commuting tuple consisting of positive operators on a Hilbert space, satisfying , we show that the spectrum of the Lüders operation: is contained in , so the only solution of the equation is the “expected” one: .

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


Passive states for essential observers
View Description Hide DescriptionThe aim of this note is to present a unified approach to the results given by Borchers and Buchholz [“Global properties of vacuum states in de Sitter space,” Ann. Inst. Henri Poincare, Sect. A.70, 23–40 (1999)] and by Buchholz and Summers [“Stable quantum systems in antide Sitter space: Causality, independence and spectral properties,” J. Math. Phys.45, 4810–4831 (2004)] which also covers examples of models not presented in these two papers (e.g., dimensional Minkowski spacetime for ). Assuming that a state is passive for an observer traveling along certain (essential) worldlines, we show that this state is invariant under the isometry group, is a temperature equilibrium state for the observer at a temperature uniquely determined by the structure constants of the Lie algebra involved, and fulfills (a variant of) the ReehSchlieder property. Also, the modular objects associated with such a state and the observable algebra of an observer are computed and a version of weak locality is examined.

On the eigensolutions of the onedimensional Duffin–Kemmer–Petiau oscillator
View Description Hide DescriptionThe onedimensional eigenfunctions and the eigenvalues of massive spin0 and spin1 particles have been found by using the Duffin–Kemmer–Petiau equation. Following Greiner [Quantum Mechanics: An introduction, 4th ed. (Springerverlag, Berlin, 2001)], we have shown that the eigensolutions in both cases are decoupled in two sets.

 GENERAL RELATIVITY AND GRAVITATION


Stationary axisymmetric solutions involving a third order equation irreducible to Painlevé transcendents
View Description Hide DescriptionWe extend the method of separation of variables, studied by Léauté and Marcilhacy [Ann. Inst. Henri Poincare, Sect. A331, 363 (1979)], to obtain transcendent solutions of the field equations for stationary axisymmetric systems. These solutions depend on transcendent functions satisfying a third order differential equation. For some solutions this equation satisfies the necessary conditions, but not sufficient, to have fixed critical points.

The principle of equivalence and cosmological metrics
View Description Hide DescriptionThis paper is concerned with the extent to which the geodesics of spacetime (that is, the experimental consequences of the principle of equivalence) determine the metric in general relativitytheory and, in particular, in Friedmann–Robertson–Walker–Lemaitre (FRWL) spacetimes. Thus it discusses projective structure in these spacetimes. The approach will be from a geometrical point of view and it is shown that if two spacetime metrics share the same (unparametrized) geodesics and one is a (generic) FRWL metric then so is the other and that each is a member of a well defined family of projectively related (FRWL) metrics. Similar techniques are then applied to study the existence and properties of symmetries of the Weyl projective tensor and projective symmetries in FRWL spacetimes.

On local equivalence problem of spacetimes with two orthogonally transitive commuting Killing fields
View Description Hide DescriptionConsidered is the problem of local equivalence of generic fourdimensional metrics possessing two commuting and orthogonally transitive Killing vector fields. A sufficient set of eight differential invariants is explicitly constructed, among them four of first order and four of second order in terms of metric coefficients. In vacuum case, the four firstorder invariants suffice to distinguish generic metrics.

Positive mass theorems for higher dimensional Lorentzian manifolds
View Description Hide DescriptionWe extend Witten’s proof [“A new proof of the positive energy theorem,” Commun. Math. Phys.80, 381–402 (1981)] of the positive mass theorem to a rigorous proof for Lorentzian manifolds of any dimension. This includes the original higherdimensional positive mass theorem proved by spinors [Parker, T. and Taubes, C., “On Witten’s proof of the positive energy theorem,” Commun. Math. Phys.84, 223–238 (1982)] for dimension 4 and [Zhang, X., “Positive mass conjecture for fivedimensional Lorentzian manifolds,” J. Math. Phys.40, 3540–3552 (1999)] for dimension 5. Stimulated by the article of Zhang [“Positive mass theorem for hypersurface in 5dimensional Lorentzian manifolds,” Commun. Anal. Geom.8, 635–652 (2000)], we weaken the spin condition on the spacelike hypersurface to require only a structure and give a modified positive mass theorem for Lorentzian manifolds for dimensions 4, 5, and 6.

 DYNAMICAL SYSTEMS


Differential Galois obstructions for integrability of homogeneous Newton equations
View Description Hide DescriptionIn this paper, we formulate necessary conditions for the integrability in the Jacobi sense of Newton equations, where and all components of are polynomial and homogeneous of the same degree . These conditions are derived from an analysis of the differential Galois group of variational equations along special particular solutions of the Newton equations. We show that, taking all admissible particular solutions, we restrict considerably the set of Newton’s equations satisfying the necessary conditions for the integrability. Moreover, we apply the obtained conditions for a detailed analysis of the Newton equations with two degrees of freedom (i.e., ). We demonstrate the strength of the obtained results analyzing general cases with . For , we have found an integrable case when the Newton equations have two polynomial first integrals and both of them are of degree 4 in the momenta and . Moreover, for an arbitrary , we found a particular family of Newton equations depending on one parameter . For an arbitrary value of , one quadratic in the momenta first integral exists. We distinguished infinitely many values of for which the system is integrable or superintegrable with additional polynomial first integrals which seemingly can be of an arbitrarily high degree with respect to the momenta.

Chaotic, fractal, and coherent solutions for a new integrable system of equations in dimensions
View Description Hide DescriptionA new integrable and nonlinear system of partial differential equations in dimensions is obtained by means of an asymptotically exact reduction method based on Fourier expansion and spatiotemporal rescaling. We find interacting coherent excitations such as the soliton, dromion, lump, ring soliton, breather, and instanton solutions. The interaction between the localized solutions are completely elastic because they pass through each other and preserve their shapes and velocities, the only change being a phase shift. Moreover, the arbitrariness of the functions included in the general solution implies that lower dimensional chaotic patterns such as chaoticchaotic patterns, periodicchaotic patterns, chaoticsoliton patterns, and chaoticdromion patterns can appear in the solution. In a similar way, fractal dromion patterns and stochastic fractal excitations also exist for appropriate choices of the initial conditions.

The vortex problem on a symmetric ellipsoid: A perturbation approach
View Description Hide DescriptionWe consider the vortex problem on an ellipsoid of revolution. Applying standard techniques of classical perturbation theory, we construct a sequence of conformal transformations from the ellipsoid into the complex plane. Using these transformations, the equations of motion for the vortex problem on the ellipsoid are written as a formal series on the eccentricity of the ellipsoid’s generating ellipse. First order equations are obtained explicitly. We show numerically that the truncated first order system for the three vortex system on the symmetric ellipsoid is nonintegrable.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


On geometric discretization of elasticity
View Description Hide DescriptionThis paper presents a geometric discretization of elasticity when the ambient space is Euclidean. This theory is built on ideas from algebraic topology, exterior calculus, and the recent developments of discrete exterior calculus. We first review some geometric ideas in continuum mechanics and show how constitutive equations of linearized elasticity, similar to those of electromagnetism, can be written in terms of a material Hodge star operator. In the discrete theory presented in this paper, instead of referring to continuum quantities, we postulate the existence of some discrete scalarvalued and vectorvalued primal and dual differential forms on a discretized solid, which is assumed to be a triangulated domain. We find the discrete governing equations by requiring energy balance invariance under timedependent rigid translations and rotations of the ambient space. There are several subtle differences between the discrete and continuous theories. For example, power of tractions in the discrete theory is written on a layer of cells with a nonzero volume. We obtain the compatibility equations of this discrete theory using tools from algebraic topology. We study a discrete Cosserat medium and obtain its governing equations. Finally, we study the geometric structure of linearized elasticity and write its governing equations in a matrix form. We show that, in addition to constitutive equations, balance of angular momentum is also metric dependent; all the other governing equations are topological.
