Volume 51, Issue 10, October 2010
 ARTICLES

 Quantum Mechanics (General and Nonrelativistic)

Localization of multidimensional Wigner distributions
View Description Hide DescriptionA well known result of P. Flandrin states that a Gaussian uniquely maximizes the integral of the Wigner distribution over every centered disk in the phase plane. While there is no difficulty in generalizing this result to higherdimensional polydisks, the generalization to balls is less obvious. In this note we provide such a generalization.

Spectral theory of semibounded Sturm–Liouville operators with local interactions on a discrete set
View Description Hide DescriptionWe study the Hamiltonians with type point interactions at the centers on the positive half line in terms of energy forms. We establish analogs of some classical results on operators with locally integrable potentials . In particular, we prove that the Hamiltonian is selfadjoint if it is lower semibounded. This result completes the previous results of Brasche [“Perturbation of Schrödinger Hamiltonians by measures—selfadjointness and semiboundedness,” J. Math. Phys.26, 621 (1985)] on lower semiboundedness. Also we prove the analog of Molchanov’s discreteness criteria, Birman’s result on stability of a continuous spectrum, and investigate discreteness of a negative spectrum. In the recent paper [Kostenko, A. and Malamud, M., “1–D Schrödinger operators with local point interactions on a discrete set,” J. Differ. Equations249, 253 (2010)], it was shown that the spectral properties of correlate with the corresponding spectral properties of a certain class of Jacobi matrices. We apply the above mentioned results to the study of spectral properties of these Jacobi matrices.

Graph approach to quantum systems
View Description Hide DescriptionUsing a graph approach to quantum systems, we show that descriptions of 3dim Kochen–Specker (KS) setups as well as descriptions of 3dim spin systems by means of Greechie diagrams (a kind of lattice) that we find in the literature are wrong. Correct lattices generated by McKayMegillPavicic (MMP) hypergraphs and Hilbert subspace equations are given. To enable future exhaustive generation of 3dim KS setups by means of our recently found stripping technique, bipartite graph generation is used to provide us with lattices with equal numbers of elements and blocks (orthogonal triples of elements)—up to 41 of them. We obtain several new results on such lattices and hypergraphs, in particular, on properties such as superposition and orthoraguesian equations.

On the low energy behavior of Regge poles
View Description Hide DescriptionWe investigate the behavior of Regge poles in the low energy limit. With the use of small argument asymptotics of the spherical Hankel functions, we show that for a finite square well potential, the associated Regge poles tend to the spectral points of the limiting selfadjoint problem. This is generalized to a compactly supported potential by applying a resolvent argument to the difference of the nonzero and zero energy wavefunctions. Furthermore, by an integral equation method we prove analogous results for a potential such that is integrable. This confirms the experimental results which show that Regge poles formed during low energy electron elasticscattering become stable bound states.

Generalized MICZKepler system, duality, polynomial, and deformed oscillator algebras
View Description Hide DescriptionWe present the quadratic algebra of the generalized MICZKepler system in threedimensional Euclidean space and its dual, the fourdimensional singular oscillator, in fourdimensional Euclidean space. We present their realization in terms of a deformed oscillatoralgebra using the Daskaloyannis construction. The structure constants are, in these cases, functions not only of the Hamiltonian but also of other integrals commuting with all generators of the quadratic algebra. We also present a new algebraic derivation of the energy spectrum of the MICZKepler system on the three sphere using a quadratic algebra. These results point out also that results and explicit formula for structure functions obtained for quadratic, cubic, and higher order polynomialalgebras in the context of twodimensional superintegrable systems may be applied to superintegrable systems in higher dimensions with and without monopoles.

Freefall in a uniform gravitational field in noncommutative quantum mechanics
View Description Hide DescriptionWe study the freefall of a quantum particle in the context of noncommutative quantum mechanics (NCQM). Assuming noncommutativity of the canonical type between the coordinates of a twodimensional configuration space, we consider a neutral particle trapped in a gravitational well and exactly solve the energy eigenvalue problem. By resorting to experimental data from the GRANIT experiment, in which the first energy levels of freely falling quantum ultracold neutrons were determined, we impose an upperbound on the noncommutativity parameter. We also investigate the time of flight of a quantum particle moving in a uniform gravitational field in NCQM. This is related to the weak equivalence principle. As we consider stationary, energy eigenstates, i.e., delocalized states, the time of flight must be measured by a quantum clock, suitably coupled to the particle. By considering the clock as a small perturbation, we solve the (stationary) scattering problem associated and show that the time of flight is equal to the classical result, when the measurement is made far from the turning point. This result is interpreted as an extension of the equivalence principle to the realm of NCQM.

Dynamical revivals in spatiotemporal evolution of driven one dimensional box
View Description Hide DescriptionWe study periodically driven one dimensional box in the presence of weak modulating force and provide analytical calculations for time of quantum revivals and explain via spatiotemporal behavior. Presently introduced analysis can be applied to calculate the time of revivals in the system subject to any periodically changing weak external driving force. Spatiotemporal behavior provides the modified numerical values of revival times which have good agreement with our analytical results. Moreover, spatiotemporal evolution shows absence of symmetry beyond a revival time, which contributes to change of slope in subsequent canals and ridges.

Harmonic oscillator in twisted Moyal plane: Eigenvalue problem and relevant properties
View Description Hide DescriptionThis paper reports on a study of a harmonic oscillator (ho) in the twisted Moyal space, in a well defined matrix basis, generated by the vector fields, which induce a dynamical star product. The usual multiplication law can be hence reproduced in the null limit. The star actions of creation and annihilation functions are explicitly computed. The ho states are infinitely degenerated with energies depending on the coordinate functions.
 Quantum Information and Computation

Distilling entanglement from arbitrary resources
View Description Hide DescriptionWe obtain the general formula for the optimal rate at which singlets can be distilled from any given noisy and arbitrarily correlated entanglement resource by means of local operations and classical communication (LOCC). Our formula, obtained by employing the quantum informationspectrum method, reduces to that derived by Devetak and Winter [Proc. R. Soc. London, Ser. A461, 207 (2005)], in the special case of an independent and identically distributed resource. The proofs rely on a oneshot version of the socalled “hashing bound,” which, in turn, provides bounds on the oneshot distillable entanglement under general LOCC.

Quantum Ustatistics
View Description Hide DescriptionThe notion of a statistic for an tuple of identical quantum systems is introduced in analogy to the classical (commutative) case: given a selfadjoint “kernel” acting on with , we define the symmetric operator with being the kernel acting on the subset of . If the systems are prepared in the product state , it is shown that the sequence of properly normalized statistics converges in moments to a linear combination of Hermite polynomials in canonical variables of a canonical commutation relation algebra defined through the quantum central limit theorem. In the special cases of nondegenerate kernels and kernels of order of 2, it is shown that the convergence holds in the stronger distribution sense. Two types of applications in quantum statistics are described: testing beyond the two simple hypotheses scenario and quantum metrology with interacting Hamiltonians.

Tensor powers of 2positive maps
View Description Hide DescriptionBy studying tensor powers of a positive map of into for finite dimensional Hilbert spaces and , we give conditions for a map , such that 2positive for all n to be nondistillable and completely positive.

Restricted numerical range: A versatile tool in the theory of quantum information
View Description Hide DescriptionNumerical range of a Hermitian operator is defined as the set of all possible expectation values of this observable among a normalized quantum state. We analyze a modification of this definition in which the expectation value is taken among a certain subset of the set of all quantum states. One considers, for instance, the set of real states, the set of product states, separable states, or the set of maximally entangled states. We show exemplary applications of these algebraic tools in the theory of quantum information: analysis of positive maps and entanglement witnesses, as well as study of the minimal output entropy of a quantum channel. Product numerical range of a unitary operator is used to solve the problem of local distinguishability of a family of two unitary gates.
 Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)

Deformations of Maxwell superalgebras and their applications
View Description Hide DescriptionWe describe the Lie algebra deformations of Maxwell superalgebra that was recently introduced as the symmetry algebra of a kappasymmetric massless superparticle in a supersymmetric constant electromagnetic backgrounds. Further we introduce the Maxwell superalgebra and present all its possible deformations. Finally, the deformed superalgebras are used to derive via a contraction procedure the complete set of Casimir operators for and Maxwell superalgebras.

On Duffin–Kemmer–Petiau particles with a mixed minimalnonminimal vector coupling and the nondegenerate boundstates for the onedimensional inversely linear background
View Description Hide DescriptionThe problem of spin0 and spin1 bosons in the background of a general mixing of minimal and nonminimal vector inversely linear potentials is explored in a unified way in the context of the Duffin–Kemmer–Petiau theory. It is shown that spin0 and spin1 bosons behave effectively in the same way. An orthogonality criterion is set up and it is used to determine uniquely the set of solutions as well as to show that evenparity solutions do not exist.

Logarithmic correlators in nonrelativistic conformal field theory
View Description Hide DescriptionWe show how logarithmic terms may arise in the correlators of fields which belong to the representation of the Schrödinger–Virasoro algebra or the affine Galilean conformal algebra (GCA). We show that in GCA, only scaling operator can have a Jordan form and rapidity cannot. We observe that in both algebras, logarithmic dependence appears along the time direction alone.

Stochastic quantization of realtime thermal field theory
View Description Hide DescriptionWe use the stochastic quantization method to obtain the free scalar propagator of a finite temperature field theory formulated in the Minkowski spacetime. First, we use the Markovian stochastic quantization approach to present the twopoint function of the theory. Second, we assume a Langevin equation with a memory kernel and a colored noise. The convergence of the Markovian and nonMarkovian stochastic processes in the asymptotic limit of the fictitious time is obtained. Our formalism can be the starting point to discuss systems at finite temperature out of equilibrium.
 General Relativity and Gravitation

Petrov type pure radiation fields of Kundt’s class
View Description Hide DescriptionWe present all Petrov type pure radiation spacetimes, with or without cosmological constant, with a shearfree, nondiverging geodesic principal null congruence. We thus completely solve the problem of aligned Petrov type pure radiation fields: either these are Robinson–Trautman spacetimes and are all explicitly known or they belong to the Kundt family, for which so far only isolated examples existed in the literature.

Iwasawa attractors
View Description Hide DescriptionStarting from the symplectic construction of the Lie algebra due to Adams, we consider an Iwasawa parametrization of the coset , which is the scalar manifold of , supergravity. Our approach, and the manifest offshell symmetry of the resulting symplectic frame, is determined by a noncompact Cartan subalgebra of the maximal subgroup of . In the absence of gauging, we utilize the explicit expression of the Lie algebra to study the origin of as scalar configuration of a BPS extremal black holeattractor. In such a framework, we highlight the action of a symmetry spanning the dyonic BPS attractors. Within a suitable supersymmetry truncation allowing for the embedding of the Reissner–Nördstrom black hole, this action is interpreted as nothing but the global symmetry of pure supergravity. Moreover, we find that the above mentioned symmetry is broken down to a discrete subgroup , implying that all BPS Iwasawa attractors are nondyonic near the origin of the scalar manifold. We can trace this phenomenon back to the fact that the Cartan subalgebra of used in our construction endows the symplectic frame with a manifest offshell covariance which is smaller than itself. Thus, the consistence of the Adams–Iwasawa symplectic basis with the action of the symmetry gives rise to the observed residual nondyonic symmetry.

On a class of global characteristic problems for the Einstein vacuum equations with small initial data
View Description Hide DescriptionWe study a class of global characteristic problems for the Einstein vacuum equations with small initial data. In a previous work [Caciotta and Nicolò, “Global characteristic problem for Einstein vacuum equations with small initial data: (I) The initial data constraints,” Journal of Hyperbolic Differential Equations (JHDE)2, 201 (2005)], denoted by Paper I, our attention was focused on prescribing the initial data satisfying the constraints imposed by the characteristic problem. In this paper we show how the global existence result can be achieved. This result is heavily based on the global results of Christodoulou and Klainerman [The Global Nonlinear Stability of the Minkowski Space, Princeton Mathematical Series Vol. 41 (Princeton University Press, Princeton, NJ, 1993)] and Klainerman and Nicolò [The Evolution Problem in General Relativity, Progress in Mathematical Physics Vol. 25 (Birkhauser, Boston, MA, 2002)].

Timedependent scattering theory for charged Dirac fields on a Reissner–Nordström black hole
View Description Hide DescriptionIn this paper, we prove a complete timedependent scattering theory for charged (massive or not) Dirac fields outside a Reissner–Nordström black hole. We shall take the point of view of observers static at infinity, well described by the Schwarzschild system of coordinates. For such observers, the exterior of a Reissner–Nordström black hole is a smooth manifold having two distinct asymptotic regions: the horizon and spacelike infinity. We first simplify the later analysis using the spherical symmetry of the Reissner–Nordström black hole and we reduce the initial dimensional evolution equation of hyperbolic type into a dimensional one. Then, we establish various propagation estimates for such fields in the same spirit as in the works by Dereziński and Gérard [Scattering Theory of Classical and Quantum NParticle Systems (SpringerVerlag, Berlin, 1997)]. We construct the asymptotic velocity operators and we show that their spectra are equal to and . This information points out the very distinct behaviors of Dirac fields near the two asymptotic regions of the black hole. As a consequence of this construction, we prove the existence and asymptotic completeness of (Dollard modified at infinity) wave operators.