Volume 52, Issue 11, November 2011
Index of content:
 ARTICLES


Quantum Mechanics (General and Nonrelativistic)

Positive quantization in the presence of a variable magnetic field
View Description Hide DescriptionStarting with a previously constructed family of coherent states, we introduce the Berezin quantization for a particle in a variable magnetic field and we show that it constitutes a strict quantization of a natural Poisson algebra. The phasespace reinterpretation involves a magnetic version of the Bargmann space and leads naturally to BerezinToeplitz operators.

Geometric spectral inversion for singular potentials
View Description Hide DescriptionThe function E = F(v) expresses the dependence of a discrete eigenvalueE of the Schrödinger Hamiltonian H = −Δ + vf(r) on the coupling parameter v. We use envelope theory to generate a functional sequence {f ^{[k]}(r)} to reconstruct f(r) from F(v) starting from a seed potential f ^{[0]}(r). In the powerlaw or log cases, the inversion can be effected analytically and is complete in just two steps. In other cases, convergence is observed numerically. To provide concrete illustrations of the inversion method it is first applied to the Hulthén potential, and it is then used to invert spectral data generated by singular potentials with shapes of the form f(r) = −a/r + b sgn(q)r ^{ q } and f(r) = −a/r + bln (r), a, b > 0. For the class of attractive central potentials with shapes f(r) = g(r)/r, with g(0) < 0 and g ^{′}(r) ⩾ 0, we prove that the groundstate energy curve F(v) determines f(r) uniquely.

Exponentially localized Wannier functions in periodic zero flux magnetic fields
View Description Hide DescriptionIn this work, we investigate conditions which ensure the existence of an exponentially localized Wannier basis for a given periodic hamiltonian. We extend previous results [Panati, G., Ann. Henri Poincare8, 995–1011 (2007)10.1007/s0002300703268] to include periodic zero flux magnetic fields which is the setting also investigated by Kuchment [J. Phys. A: Math. Theor.42, 025203 (2009)10.1088/17518113/42/2/025203]. The new notion of magnetic symmetry plays a crucial rôle; to a large class of symmetries for a nonmagnetic system, one can associate “magnetic” symmetries of the related magnetic system. Observing that the existence of an exponentially localized Wannier basis is equivalent to the triviality of the socalled Bloch bundle, a rank m hermitian vector bundle over the Brillouin zone, we prove that magnetic timereversal symmetry is sufficient to ensure the triviality of the Bloch bundle in spatial dimensiond = 1, 2, 3. For d = 4, an exponentially localized Wannier basis exists provided that the trace per unit volume of a suitable function of the Fermi projection vanishes. For d > 4 and d ⩽ 2m (stable rank regime) only the exponential localization of a subset of Wannier functions is shown; this improves part of the analysis of Kuchment [J. Phys. A: Math. Theor.42, 025203 (2009)10.1088/17518113/42/2/025203]. Finally, for d > 4 and d > 2m (unstable rank regime) we show that the mere analysis of Chern classes does not suffice in order to prove triviality and thus exponential localization.

Euclidean Jordan algebras, hidden actions, and JKepler problems
View Description Hide DescriptionFor a simple euclidean Jordan algebra, let be its conformal algebra, be the manifold consisting of its semipositive rankone elements, be the space of complexvalued smooth functions on . An explicit action of on , referred to as the hidden action of on , is exhibited. This hidden action turns out to be mathematically responsible for the existence of the Kepler problem and its recently discovered vast generalizations, referred to as JKepler problems. The JKepler problems are then reconstructed and reexamined in terms of the unified language of euclidean Jordan algebras. As a result, for a simple euclidean Jordan algebra, the minimal representation of its conformal group can be realized either as the Hilbert space of bound states for its JKepler problem or as , where vol is the volume form on and r is the inner product of with the identity element of the Jordan algebra.

New approximate method to solve the Schrödinger equation with a WoodsSaxonlike potential
View Description Hide DescriptionWe presented a new method to solve Schrödinger equations especially for two special kinds of potentials, which are named the first and second kind of WoodsSaxonlike potentials in this paper. The WoodsSaxonlike potential characterized by a rapid increase occurred at the system's boundary varies slowly inside and quickly becomes a constant potential outside the system. The first (second) kind of WoodsSaxonlike potentials is finite (divergent) at the origin. By using an elaborately constructed multistep potential to approximate the WoodsSaxonlike potential, we can obtain its approximate energy levels and piecewise analytical wave functions with high accuracy. To test our method, we solved the Schrödinger equations of three systems atomic nuclei ^{208}Pb, hydrogen atoms, and sodiumnanospheres. We found that our method works quite well and is superior to conventional numerical methods for the situation of WoodsSaxonlike potentials. Besides being able to obtain approximate piecewise analytical wave functions, our method has two explicit advantages (a) the absolute error of energy levels is controlled by the number of the potential steps of the multistep approximate potential, and (b) the potential is not necessary to have an analytical expression.

Quantum Information and Computation

Continuity bounds on the quantum relative entropy — II
View Description Hide DescriptionThe quantum relative entropy is frequently used as a distance measure between two quantum states, and inequalities relating it to other distance measures are important mathematical tools in many areas of quantum informationtheory. We have derived many such inequalities in previous work. The present paper is a followup on this, and provides a sharp upper bound on the relative entropy in terms of the trace norm distance and of the smallest eigenvalues of both states concerned. The result obtained here is more general than the corresponding one from our previous work. As a corollary, we obtain a sharp upper bound on the regularised relative entropy introduced by Lendi, Farhadmotamed, and van Wonderen.

Average output entropy for quantum channels
View Description Hide DescriptionWe study the regularized average Renyi output entropy of quantum channels. This quantity gives information about the average noisiness of the channel output arising from a typical, highly entangled input state in the limit of infinite dimensions. We find a closed expression for , a quantity which we conjecture to be equal to . We find an explicit form for for some entanglementbreaking channels and also for the qubit depolarizing channel Δ_{λ} as a function of the parameter λ. We prove equality of the two quantities in some cases, in particular, we conclude that for Δ_{λ} both are nonanalytic functions of the variable λ.

Gröbner bases for finitetemperature quantum computing and their complexity
View Description Hide DescriptionFollowing the recent approach of using order domains to construct Gröbner bases from general projective varieties, we examine the parity and timereversal arguments relating to the Wightman axioms of quantum field theory and propose that the definition of associativity in these axioms should be introduced a posteriori to the cluster property in order to generalize the anyon conjecture for quantum computing to indefinite metrics. We then show that this modification, which we define via ideal quotients, does not admit a faithful representation of the Braid group, because the generalized twisted inner automorphisms that we use to reintroduce associativity are only parity invariant for the prime spectra of the exterior algebra. We then use a coordinate prescription for the quantum deformations of toric varieties to show how a faithful representation of the Braid group can be reconstructed and argue that for a degree reverse lexicographic (monomial) ordered Gröbner basis, the complexity class of this problem is bounded quantum polynomial.

General Relativity and Gravitation

Maximal slicings in spherical symmetry: Local existence and construction
View Description Hide DescriptionWe show that any spherically symmetric spacetime locally admits a maximal spacelike slicing and we give a procedure allowing its construction. The designed construction procedure is based on purely geometrical arguments and, in practice, leads to solve a decoupled system of firstorder quasilinear partial differential equations. We have explicitly built up maximal foliations in Minkowski and Friedmann spacetimes. Our approach admits further generalizations and efficient computational implementation. As byproduct, we suggest some applications of our work in the task of calibrating numerical relativity complex codes, usually written in Cartesian coordinates.

Dirac spinors in BianchiI f(R)cosmology with torsion
View Description Hide DescriptionWe study Dirac spinors in Bianchi typeI cosmological models, within the framework of torsional f(R)gravity. We find four types of results: the resulting dynamic behavior of the universe depends on the particular choice of function f(R); some f(R) models do not isotropize and have no Einstein limit, so that they have no physical significance, whereas for other f(R) models isotropization and Einsteinization occur, and so they are physically acceptable, suggesting that phenomenological arguments may select f(R) models that are physically meaningful; the singularity problem can be avoided, due to the presence of torsion; the general conservation laws holding for f(R)gravity with torsion ensure the preservation of the Hamiltonian constraint, so proving that the initial value problem is wellformulated for these models.

Sphere orderings representation of spacetime
View Description Hide DescriptionA spacetime may be regarded as a partially ordered set by causal relations. R. J. Low considered a continuous sphere order representation of a spacetime and obtained some interesting results. However, some important questions about the conditions of continuity of this representation are still left unanswered. In this paper, we considered the problem of continuity and presented a partial solution for the set of all discontinuity points of a sphere order representation and constructed an example of a spacetime which could not admit a sphere order representation with a special dimension. We also showed that, for a poset, the definition of the 2sphere ordering dimension is equivalent to the definition of the 2Euclidian dimension.

Global hyperbolicity is stable in the interval topology
View Description Hide DescriptionWe prove that global hyperbolicity is stable in the interval topology on the spacetime metrics. We also prove that every globally hyperbolic spacetime admits a Cauchy hypersurface which remains Cauchy under small perturbations of the spacetime metric. Moreover, we prove that if the spacetime admits a complete timelike Killing field, then the light cones can be widened preserving both global hyperbolicity and the Killing property of the field.

Dynamical Systems

Global dynamics of stationary solutions of the extended Fisher–Kolmogorov equation
View Description Hide DescriptionIn this paper we study the fourth order differential equation, which arises from the study of stationary solutions of the Extended Fisher–Kolmogorov equation. Denoting this equation becomes equivalent to the polynomial system with and As usual, the dot denotes the derivative with respect to the time t. Since the system has a first integral we can reduce our analysis to a family of systems on We provide the global phase portrait of these systems in the Poincaré ball (i.e., in the compactification of with the sphere of the infinity).

Nonintegrability of a class of Hamiltonian systems
View Description Hide DescriptionIn this paper, using the MoralesRamis theory, we will give some new nonintegrability results for a class of famous Hamiltonian models.

Conditional stability theorem for the one dimensional KleinGordon equation
View Description Hide DescriptionThe paper addresses the conditional nonlinear stability of the steady state solutions of the onedimensional KleinGordon equation for large time. We explicitly construct the centerstable manifold for the steady state solutions using the modulation method of Soffer and Weinstein and Strichartz type estimates. The main difficulty in the onedimensional case is that the required decay of the KleinGordon semigroup does not follow from Strichartz estimates alone. We resolve this issue by proving an additional weighted decay estimate and further refinement of the function spaces, which allows us to close the argument in spaces with very little time decay.

Dynamical systems defining Jacobi's ϑconstants
View Description Hide DescriptionWe propose a system of equations that defines Weierstrass–Jacobi's eta and thetaconstant series in a differentially closed way. This system is shown to have a direct relationship to a littleknown dynamical system obtained by Jacobi. The classically known differential equations by Darboux–Halphen, Chazy, and Ramanujan are the differential consequences or reductions of these systems. The proposed system is shown to admit the Lagrangian, Hamiltonian, and Nambu formulations. We explicitly construct a pencil of nonlinear Poisson brackets and complete set of involutive conserved quantities. As byproducts of the theory, we exemplify conserved quantities for the Ramamani dynamical system and quadratic system of Halphen–Brioschi.

Statistical Physics

Binary jumps in continuum. II. Nonequilibrium process and a Vlasovtype scaling limit
View Description Hide DescriptionLet Γ denote the space of all locally finite subsets (configurations) in . A stochastic dynamics of binary jumps in continuum is a Markov process on Γ in which pairs of particles simultaneously hop over . We discuss a nonequilibrium dynamics of binary jumps. We prove the existence of an evolution of correlation functions on a finite time interval. We also show that a Vlasovtype mesoscopic scaling for such a dynamics leads to a generalized Boltzmann nonlinear equation for the particle density.

Absence of orderings in Hubbard models
View Description Hide DescriptionIn this paper, an absence of certain orderings in multiparameter family of Hubbard models is proved. A technical tool is a reflection positivity, which allows to establish upper bounds on certain susceptibilities. These bounds in turn imply an absence of magnetic, charge, and superconducting orderings. The models considered are extended Hubbard models with charge and spin interactions, correlation hopping, and pair hopping.

Counting spanning trees in selfsimilar networks by evaluating determinants
View Description Hide DescriptionSpanning trees are relevant to various aspects of networks. Generally, the number of spanning trees in a network can be obtained by computing a related determinant of the Laplacian matrix of the network. However, for a large generic network, evaluating the relevant determinant is computationally intractable. In this paper, we develop a fairly generic technique for computing determinants corresponding to selfsimilar networks, thereby providing a method to determine the numbers of spanning trees in networks exhibiting selfsimilarity. We describe the computation process with a family of networks, called (x, y)flowers, which display rich behavior as observed in a large variety of real systems. The enumeration of spanning trees is based on the relationship between the determinants of submatrices of the Laplacian matrix corresponding to the (x, y)flowers at different generations and is devoid of the direct laborious computation of determinants. Using the proposed method, we derive analytically the exact number of spanning trees in the (x, y)flowers, on the basis of which we also obtain the entropies of the spanning trees in these networks. Moreover, to illustrate the universality of our technique, we apply it to some other selfsimilar networks with distinct degree distributions, and obtain explicit solutions to the numbers of spanning trees and their entropies. Finally, we compare our results for networks with the same average degree but different structural properties, such as degree distribution and fractal dimension, and uncover the effect of these topological features on the number of spanning trees.

Methods of Mathematical Physics

On the Hurwitz zeta function of imaginary second argument
View Description Hide DescriptionIn this work, we exploit Jonquière's formula relating the Hurwitz zeta function to a linear combination of polylogarithmic functions in order to evaluate the real and imaginary part of ζ_{ H }(s, ia) and its first derivative with respect to the first argument s. In particular, we obtain expressions for the real and imaginary party of ζ_{ H }(s, ia) and its derivative for s = m with involving simpler transcendental functions. We apply these results to the computation of the imaginary part of the oneloop effective action for massive scalar fields under the influence of a strong electric field in higher dimensional Minkowski spacetime.
