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Mar 2008

Volume 49, Issue 3, Articles (03xxxx)

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Solitary wave dynamics in time-dependent potentials

Walid K. Abou Salem

J. Math. Phys. 49, 032101 (2008); http://dx.doi.org/10.1063/1.2837429 (29 pages) | Cited 7 times

Online Publication Date: 6 March 2008

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The long time dynamics of solitary wave solutions of the nonlinear Schrödinger equation in time-dependent external potentials is rigorously studied. To set the stage, the well-posedness of the Cauchy problem for a generalized nonautonomous nonlinear Schrödinger equation with time-dependent nonlinearities and potential is established. Afterward, the dynamics of NLS solitary waves in time-dependent potentials is studied. It is shown that in the space-adiabatic regime where the external potential varies slowly in space compared to the size of the soliton, the dynamics of the center of the soliton is described by Hamilton’s equations, plus terms due to radiation damping. Finally, two physical applications are discussed: the first is adiabatic transportation of solitons and the second is the Mathieu instability of trapped solitons due to time-periodic perturbations.
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05.45.Yv Solitons
03.65.Ge Solutions of wave equations: bound states
02.30.-f Function theory, analysis

The divine clockwork: Bohr’s correspondence principle and Nelson’s stochastic mechanics for the atomic elliptic state

Richard Durran, Andrew Neate, and Aubrey Truman

J. Math. Phys. 49, 032102 (2008); http://dx.doi.org/10.1063/1.2837434 (30 pages) | Cited 3 times

Online Publication Date: 6 March 2008

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We consider the Bohr correspondence limit of the Schrödinger wave function for an atomic elliptic state. We analyze this limit in the context of Nelson’s stochastic mechanics, exposing an underlying deterministic dynamical system in which trajectories converge to Keplerian motion on an ellipse. This solves the long standing problem of obtaining Kepler’s laws of planetary motion in a quantum mechanical setting. In this quantum mechanical setting, local mild instabilities occur in the Keplerian orbit for eccentricities greater than math which do not occur classically.
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03.65.Pm Relativistic wave equations
03.65.Ca Formalism
02.50.Ey Stochastic processes

Rotations of occupied invariant subspaces in self-consistent field calculations

Emanuel H. Rubensson, Elias Rudberg, and Paweł Sałek

J. Math. Phys. 49, 032103 (2008); http://dx.doi.org/10.1063/1.2884588 (9 pages) | Cited 7 times

Online Publication Date: 6 March 2008

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In this article, the self-consistent field (SCF) procedure as used in Hartree–Fock and Kohn–Sham calculations is viewed as a sequence of rotations of the so-called occupied invariant subspace of the potential and density matrices. Computational approximations are characterized as erroneous rotations of this subspace. Differences between subspaces are measured and controlled by the canonical angles between them. With this approach, a first step is taken toward a method where errors from computational approximations are rigorously controlled and threshold values are directly related to the accuracy of the current trial density, thus eliminating the use of ad hoc threshold values. Then, the use of computational resources can be kept down as much as possible without impairment of the SCF convergence.
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31.15.xr Self-consistent-field methods

The structure of classical extensions of quantum probability theory

Werner Stulpe and Paul Busch

J. Math. Phys. 49, 032104 (2008); http://dx.doi.org/10.1063/1.2884581 (22 pages) | Cited 7 times

Online Publication Date: 7 March 2008

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On the basis of a suggestive definition of a classical extension of quantum mechanics in terms of statistical models, we prove that every such classical extension is essentially given by the so-called Misra–Bugajski reduction map. We consider how this map enables one to understand quantum mechanics as a reduced classical statistical theory on the projective Hilbert space as phase space and discuss features of the induced hidden-variable model. Moreover, some relevant technical results on the topology and Borel structure of the projective Hilbert space are reviewed.
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03.65.Ta Foundations of quantum mechanics; measurement theory

Quantum reference frames and the classification of rotationally invariant maps

J.-C. Boileau, L. Sheridan, M. Laforest, and S. D. Bartlett

J. Math. Phys. 49, 032105 (2008); http://dx.doi.org/10.1063/1.2884583 (19 pages) | Cited 6 times

Online Publication Date: 13 March 2008

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We give a convenient representation for any map that is covariant with respect to an irreducible representation of SU(2), and use this representation to analyze the evolution of a quantum directional reference frame when it is exploited as a resource for performing quantum operations. We introduce the moments of a quantum reference frame, which serve as a complete description of its properties as a frame, and investigate how many times a quantum directional reference frame represented by a spin-j system can be used to perform a certain quantum operation with a given probability of success. We provide a considerable generalization of previous results on the degradation of a reference frame, from which follows a classification of the dynamics of spin-j system under the repeated action of any covariant map with respect to SU(2).
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03.65.Fd Algebraic methods
02.20.Qs General properties, structure, and representation of Lie groups

Spectral properties of the two-photon absorption and emission process

R. Carbone, F. Fagnola, J. C. García, and R. Quezada

J. Math. Phys. 49, 032106 (2008); http://dx.doi.org/10.1063/1.2890700 (18 pages)

Online Publication Date: 13 March 2008

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The quantum Markov semigroup of the two-photon absorption and emission process has two extremal normal invariant states. Starting from an arbitrary initial state it converges toward some convex combination of these states as time goes to infinity (approach to equilibrium). We compute the exact exponential rate of this convergence showing that it depends only on the emission rates. Moreover, we show that off-diagonal matrix elements of any initial state go to zero with an exponential rate which is smaller than the exponential rate of convergence of the diagonal part. In other words quantum features of a state survive longer than the relaxation time of its classical part.
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42.50.Gy Effects of atomic coherence on propagation, absorption, and amplification of light; electromagnetically induced transparency and absorption

Approximating a wavefunction as an unconstrained sum of Slater determinants

Gregory Beylkin, Martin J. Mohlenkamp, and Fernando Pérez

J. Math. Phys. 49, 032107 (2008); http://dx.doi.org/10.1063/1.2873123 (28 pages) | Cited 8 times

Online Publication Date: 14 March 2008

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The wavefunction for the multiparticle Schrödinger equation is a function of many variables and satisfies an antisymmetry condition, so it is natural to approximate it as a sum of Slater determinants. Many current methods do so, but they impose additional structural constraints on the determinants, such as orthogonality between orbitals or an excitation pattern. We present a method without any such constraints, by which we hope to obtain much more efficient expansions and insight into the inherent structure of the wavefunction. We use an integral formulation of the problem, a Green’s function iteration, and a fitting procedure based on the computational paradigm of separated representations. The core procedure is the construction and solution of a matrix-integral system derived from antisymmetric inner products involving the potential operators. We show how to construct and solve this system with computational complexity competitive with current methods.
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03.65.Ge Solutions of wave equations: bound states
03.65.Db Functional analytical methods
03.65.Fd Algebraic methods
02.60.-x Numerical approximation and analysis
02.30.Mv Approximations and expansions
02.30.Rz Integral equations
02.10.Yn Matrix theory

Solutions for a Schrödinger equation with a nonlocal term

E. K. Lenzi, B. F. de Oliveira, L. R. da Silva, and L. R. Evangelista

J. Math. Phys. 49, 032108 (2008); http://dx.doi.org/10.1063/1.2842069 (8 pages) | Cited 5 times

Online Publication Date: 14 March 2008

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We obtain time dependent solutions for a Schröndiger equation in the presence of a nonlocal term by using the Green function approach. These solutions are compared with recent results obtained for the fractional Schrödinger equation as well as for the usual one. The nonlocal term incorporated in the Schrödinger equation may also be related to the spatial and time fractional derivative and introduces different regimes of spreading of the solution with the time evolution.
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03.65.Ge Solutions of wave equations: bound states
03.65.Db Functional analytical methods

Fermionic quasifree states and maps in information theory

B. Dierckx, M. Fannes, and M. Pogorzelska

J. Math. Phys. 49, 032109 (2008); http://dx.doi.org/10.1063/1.2841326 (18 pages) | Cited 4 times

Online Publication Date: 19 March 2008

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This paper and the results therein are geared toward building a basic toolbox for calculations in quantum information theory of quasifree fermionic systems. Various entropy and relative entropy measures are discussed. The main emphasis is on completely positive quasifree maps. The set of quasifree affine maps on the state space is determined and fully characterized in terms of operations on one-particle subspaces. For a subclass of trace-preserving completely positive maps and for their duals, Choi matrices and Jamiolkowski states are discussed.
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03.67.-a Quantum information
03.65.Fd Algebraic methods
05.30.Fk Fermion systems and electron gas
02.10.Yn Matrix theory

Supersymmetric associated vector coherent states and generalized Landau levels arising from two-dimensional supersymmetry

S. Twareque Ali and F. Bagarello

J. Math. Phys. 49, 032110 (2008); http://dx.doi.org/10.1063/1.2898117 (17 pages) | Cited 9 times

Online Publication Date: 24 March 2008

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We describe a method for constructing vector coherent states for quantum supersymmetric partner Hamiltonians. The method is then applied to such partner Hamiltonians arising from a generalization of the fractional quantum Hall effect. Explicit examples are worked out.
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03.65.Ta Foundations of quantum mechanics; measurement theory
03.65.Fd Algebraic methods
02.10.Ud Linear algebra

Hiatus perturbation for a singular Schrödinger operator with an interaction supported by a curve in math3

P. Exner and S. Kondej

J. Math. Phys. 49, 032111 (2008); http://dx.doi.org/10.1063/1.2845419 (19 pages) | Cited 1 time

Online Publication Date: 27 March 2008

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We consider Schrödinger operators in L2(math3) with a singular interaction supported by a finite curve Γ. We present a proper definition of the operators and study their properties, in particular, we show that the discrete spectrum can be empty if Γ is short enough. If it is not the case, we investigate properties of the eigenvalues in the situation when the curve has a hiatus of length 2ϵ. We derive an asymptotic expansion with the leading term which a multiple of ϵ ln ϵ.
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03.65.Ge Solutions of wave equations: bound states

Error exponents in hypothesis testing for correlated states on a spin chain

Fumio Hiai, Milán Mosonyi, and Tomohiro Ogawa

J. Math. Phys. 49, 032112 (2008); http://dx.doi.org/10.1063/1.2872276 (22 pages) | Cited 6 times

Online Publication Date: 27 March 2008

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We study various error exponents in a binary hypothesis testing problem and extend recent results on the quantum Chernoff and Hoeffding bounds for product states to a setting when both the null hypothesis and the alternative hypothesis can be correlated states on a spin chain. Our results apply to states satisfying a certain factorization property; typical examples are the global Gibbs states of translation-invariant finite-range interactions as well as certain finitely correlated states.
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03.65.Fd Algebraic methods
02.10.Ud Linear algebra
02.50.Cw Probability theory

Geometry of sets of quantum maps: A generic positive map acting on a high-dimensional system is not completely positive

Stanisław J. Szarek, Elisabeth Werner, and Karol Życzkowski

J. Math. Phys. 49, 032113 (2008); http://dx.doi.org/10.1063/1.2841325 (21 pages) | Cited 3 times

Online Publication Date: 27 March 2008

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We investigate the set (a) of positive, trace preserving maps acting on density matrices of size N and a sequence of its nested subsets: the sets of maps which are (b) decomposable, (c) completely positive, and (d) extended by identity impose positive partial transpose and (e) are superpositive. Working with the Hilbert–Schmidt (Euclidean) measure, we derive tight explicit two-sided bounds for the volumes of all five sets. A sample consequence is the fact that, as N increases, a generic positive map becomes not decomposable and, a fortiori, not completely positive. Due to the Jamiołkowski isomorphism, the results obtained for quantum maps are closely connected to similar relations between the volume of the set of quantum states and the volumes of its subsets (such as states with positive partial transpose or separable states) or supersets. Our approach depends on the systematic use of duality to derive quantitative estimates and on various tools of classical convexity, high-dimensional probability, and geometry of Banach spaces, some of which are not standard.
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03.65.Ta Foundations of quantum mechanics; measurement theory
03.65.Fd Algebraic methods
02.40.-k Geometry, differential geometry, and topology
02.10.Yn Matrix theory

Maps on states preserving the relative entropy

Lajos Molnár

J. Math. Phys. 49, 032114 (2008); http://dx.doi.org/10.1063/1.2898693 (4 pages) | Cited 3 times

Online Publication Date: 31 March 2008

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Let H be a finite dimensional Hilbert space. The aim of this short note is to prove that every bijective map on the space S(H) of all density operators on H which preserves the relative entropy is of the form ϕ(ρ) = UρU* with some unitary or antiunitary operator U on H.
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02.10.Ud Linear algebra
03.65.Fd Algebraic methods
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On the Poisson-Lie T-plurality of boundary conditions

Cecilia Albertsson, Ladislav Hlavatý, and Libor Šnobl

J. Math. Phys. 49, 032301 (2008); http://dx.doi.org/10.1063/1.2832622 (23 pages) | Cited 2 times

Online Publication Date: 6 March 2008

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Conditions for the gluing matrix defining consistent boundary conditions of two-dimensional nonlinear σ-models are analyzed and reformulated. Transformation properties of the right-invariant fields under the Poisson-Lie T-plurality are used to derive a formula for the transformation of the boundary conditions. Examples of transformation of D-branes in two and three dimensions are presented. We investigate obstacles arising in this procedure and propose possible solutions.
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02.30.Jr Partial differential equations
02.60.Lj Ordinary and partial differential equations; boundary value problems
11.25.Uv D branes

Numerical Calabi–Yau metrics

Michael R. Douglas, Robert L. Karp, Sergio Lukic, and René Reinbacher

J. Math. Phys. 49, 032302 (2008); http://dx.doi.org/10.1063/1.2888403 (19 pages) | Cited 7 times

Online Publication Date: 14 March 2008

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We develop numerical methods for approximating Ricci flat metrics on Calabi–Yau hypersurfaces in projective spaces. Our approach is based on finding balanced metrics and builds on recent theoretical work by Donaldson. We illustrate our methods in detail for a one parameter family of quintics. We also suggest several ways to extend our results.
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11.25.Wx String and brane phenomenology
02.60.-x Numerical approximation and analysis

Topological photon

S. C. Tiwari

J. Math. Phys. 49, 032303 (2008); http://dx.doi.org/10.1063/1.2883828 (7 pages) | Cited 2 times

Online Publication Date: 21 March 2008

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We associate intrinsic energy equal to hν/2 with the spin angular momentum of photon, and propose a topological model based on orbifold in space and tifold in time as topological obstructions. The model is substantiated using vector wavefield disclinations. The physical photon is suggested to be a particlelike topological photon and a propagating wave such that the energy hν of photon is equally divided between spin energy and translational energy, corresponding to linear momentum of hν/c. The enigma of wave-particle duality finds natural resolution, and the proposed model gives new insights into the phenomena of interference and emission of radiation.
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02.40.Pc General topology
42.50.-p Quantum optics

On the regularized fermionic projector of the vacuum

Felix Finster

J. Math. Phys. 49, 032304 (2008); http://dx.doi.org/10.1063/1.2888187 (60 pages) | Cited 2 times

Online Publication Date: 24 March 2008

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We construct families of fermionic projectors with spherically symmetric regularization, which satisfy the condition of a distributional MP-product. The method is to analyze regularization tails with a power law or logarithmic scaling in composite expressions in the fermionic projector. The resulting regularizations break the Lorentz symmetry and give rise to a multilayer structure of the fermionic projector near the light cone. Furthermore, we construct regularizations which go beyond the distributional MP-product in that they yield additional distributional contributions supported at the origin. The remaining freedom for the regularization parameters and the consequences for the normalization of the fermionic states are discussed.
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11.10.Gh Renormalization
11.30.Cp Lorentz and Poincaré invariance
11.30.Qc Spontaneous and radiative symmetry breaking
11.10.Jj Asymptotic problems and properties

Heat kernel estimates and spectral properties of a pseudorelativistic operator with magnetic field

D. H. Jakubassa-Amundsen

J. Math. Phys. 49, 032305 (2008); http://dx.doi.org/10.1063/1.2844615 (22 pages) | Cited 2 times

Online Publication Date: 26 March 2008

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Based on the Mehler heat kernel of the Schrödinger operator for a free electron in a constant magnetic field, an estimate for the kernel of EA = ∣α(p−eA)+βm∣ is derived, where EA represents the kinetic energy of a Dirac electron within the pseudorelativistic no-pair Brown–Ravenhall model. This estimate is used to provide the bottom of the essential spectrum for the two-particle Brown–Ravenhall operator, describing the motion of the electrons in a central Coulomb field and a constant magnetic field, if the central charge is restricted to Z ⩽ 86.
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03.65.Pm Relativistic wave equations
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“Massless” vector field in de Sitter universe

T. Garidi, J.-P. Gazeau, S. Rouhani, and M. V. Takook

J. Math. Phys. 49, 032501 (2008); http://dx.doi.org/10.1063/1.2841327 (25 pages) | Cited 7 times

Online Publication Date: 19 March 2008

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We proceed to the quantization of the massless vector field in the de Sitter (dS) space. This work is the natural continuation of a previous article devoted to the quantization of the dS massive vector field [ J. P. Gazeau and M. V. Takook, J. Math. Phys. 41, 5920 (2000) ; T. Garidi et al., ibid. 43, 6379 (2002). ] The term “massless” is used by reference to conformal invariance and propagation on the dS lightcone whereas “massive” refers to those dS fields which unambiguously contract to Minkowskian massive fields at zero curvature. Due to the combined occurrences of gauge invariance and indefinite metric, the covariant quantization of the massless vector field requires an indecomposable representation of the de Sitter group. We work with the gauge fixing corresponding to the simplest Gupta–Bleuler structure. The field operator is defined with the help of coordinate-independent de Sitter waves (the modes). The latter are simple to manipulate and most adapted to group theoretical approaches. The physical states characterized by the divergencelessness condition are, for instance, easy to identify. The whole construction is based on analyticity requirements in the complexified pseudo-Riemannian manifold for the modes and the two-point function.
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98.80.Cq Particle-theory and field-theory models of the early Universe (including cosmic pancakes, cosmic strings, chaotic phenomena, inflationary universe, etc.)
11.15.-q Gauge field theories

Bianchi type-I cosmological model for perfect fluid distribution in Lyra geometry

Raj Bali and Naresh Kumar Chandnani

J. Math. Phys. 49, 032502 (2008); http://dx.doi.org/10.1063/1.2898477 (8 pages) | Cited 12 times

Online Publication Date: 21 March 2008

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In this paper, we have investigated Bianchi type-I cosmological model with time dependent gauge function β for perfect fluid distribution within the framework of Lyra geometry. To get the deterministic model of the universe, we have assumed that eigenvalue (σ11) of shear tensor (σij) is proportional to the expansion (θ). This leads to A = (BC)n, where A,B,C are metric potentials. The physical and geometrical aspects of the model and singularities in the model are also discussed.
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98.80.Jk Mathematical and relativistic aspects of cosmology
04.20.Jb Exact solutions
95.30.Sf Relativity and gravitation
02.40.-k Geometry, differential geometry, and topology

Topological higher gauge theory: From BF to BFCG theory

F. Girelli, H. Pfeiffer, and E. M. Popescu

J. Math. Phys. 49, 032503 (2008); http://dx.doi.org/10.1063/1.2888764 (17 pages) | Cited 1 time

Online Publication Date: 25 March 2008

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We study generalizations of three- and four-dimensional BF theories in the context of higher gauge theory. First, we construct topological higher gauge theories as discrete state sum models and explain how they are related to the state sums of Yetter, Mackaay, and Porter. Under certain conditions, we can present their corresponding continuum counterparts in terms of classical Lagrangians. We then explain that two of these models are already familiar from the literature: the ΣΦEA model of three-dimensional gravity coupled to topological matter and also a four-dimensional model of BF theory coupled to topological matter.
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11.15.-q Gauge field theories
02.40.Pc General topology
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Hyperincursive discrete harmonic oscillator

Adel F. Antippa and Daniel M. Dubois

J. Math. Phys. 49, 032701 (2008); http://dx.doi.org/10.1063/1.2890383 (6 pages)

Online Publication Date: 21 March 2008

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The hyperincursive algorithm for the discrete harmonic oscillator is perfectly stable and energy conserving. By identifying the natural parameters of the system, we transform the algorithm into a normal formalism based on dynamic equations of motion. We find that the simultaneous difference equations of motion are complex, that the natural parameters are classical analogs of the quantum mechanical creation and annihilation operators, and that the solution is of utmost simplicity. The methodology is applicable to any dynamical system, has conceptual importance for discrete physics, and practical utility for numerical simulations.
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45.05.+x General theory of classical mechanics of discrete systems
45.20.-d Formalisms in classical mechanics

Darboux integrability and algebraic invariant surfaces for the Rikitake system

Jaume Llibre and Clàudia Valls

J. Math. Phys. 49, 032702 (2008); http://dx.doi.org/10.1063/1.2897983 (17 pages) | Cited 2 times

Online Publication Date: 25 March 2008

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In this paper, we study the Darboux integrability of the Rikitake system math = −μx+yz, math = −μy+x(z−a), math = 1−xy. More precisely, we characterize all the invariant algebraic surfaces, the exponential factors, and the polynomial, rational, and Darboux first integrals of this system according to the values of its parameters a and μ.
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02.30.-f Function theory, analysis
02.10.De Algebraic structures and number theory

The harmonic oscillator on Riemannian and Lorentzian configuration spaces of constant curvature

José F. Cariñena, Manuel F. Rañada, and Mariano Santander

J. Math. Phys. 49, 032703 (2008); http://dx.doi.org/10.1063/1.2840463 (27 pages) | Cited 1 time

Online Publication Date: 25 March 2008

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The harmonic oscillator as a distinguished dynamical system can be defined not only on the Euclidean plane but also on the sphere and on the hyperbolic plane, and more generally on any configuration space with constant curvature and metric of any signature, either Riemannian (definite positive) or Lorentzian (indefinite). In this paper we study the main properties of these “curved” harmonic oscillators simultaneously on any such configuration space, using a Cayley–Klein (CK)-type approach, with two free parameters κ1,κ2 which altogether correspond to the possible values for curvature and signature type: the generic Riemannian and Lorentzian spaces of constant curvature (sphere S2, hyperbolic plane H2, AntiDeSitter sphere AdS1+1, and DeSitter sphere dS1+1) appear in this family, with Euclidean and Minkowski spaces as flat particular cases. We solve the equations of motion for the curved harmonic oscillator and obtain explicit expressions for the orbits by using three different methods: by direct integration, by obtaining the general CK version of Binet’s equation, and finally as a consequence of its superintegrable character. The orbits are conics with center at the potential origin on any CK space, thereby extending this well known Euclidean property to any constant curvature configuration space. The final part of the article, that has a more geometric character, presents pertinent results of the theory of conics on spaces of constant curvature.
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45.20.Jj Lagrangian and Hamiltonian mechanics
02.30.Hq Ordinary differential equations
02.40.Ky Riemannian geometries
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