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Mar 2013

Volume 54, Issue 3, Articles (03xxxx)

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J. Math. Phys. 54, 033512 (2013); http://dx.doi.org/10.1063/1.4794514 (20 pages)

Andrew Neate and Aubrey Truman
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back to top Partial Differential Equations

Ground state solutions for nonlinear fractional Schrödinger equations in mathN

Simone Secchi

J. Math. Phys. 54, 031501 (2013); http://dx.doi.org/10.1063/1.4793990 (17 pages)

Online Publication Date: 6 March 2013

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We construct solutions to a class of Schrödinger equations involving the fractional Laplacian. Our approach is variational in nature, and based on minimization on the Nehari manifold.
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03.65.Ge Solutions of wave equations: bound states
02.30.Hq Ordinary differential equations
02.30.Xx Calculus of variations

Newtonian limit and trend to equilibrium for the relativistic Fokker-Planck equation

José Antonio Alcántara Félix and Simone Calogero

J. Math. Phys. 54, 031502 (2013); http://dx.doi.org/10.1063/1.4793991 (9 pages)

Online Publication Date: 8 March 2013

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The relativistic Fokker-Planck equation, in which the speed of light c appears as a parameter, is considered. It is shown that in the limit c → ∞ its solutions converge in L1 to solutions of the non-relativistic Fokker-Planck equation, uniformly in compact intervals of time. Moreover in the case of spatially homogeneous solutions, and provided the temperature of the thermal bath is sufficiently small, exponential trend to equilibrium in L1 is established. The dependence of the rate of convergence on the speed of light is estimated. Finally, it is proved that exponential convergence to equilibrium for all temperatures holds in a weighted L2 norm.
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05.60.-k Transport processes
02.30.-f Function theory, analysis
02.50.-r Probability theory, stochastic processes, and statistics

Low-regularity solutions of the periodic modified two-component Camassa-Holm equation

Li-meng Xia, Lixin Tian, and Caixia Shen

J. Math. Phys. 54, 031503 (2013); http://dx.doi.org/10.1063/1.4794284 (10 pages)

Online Publication Date: 8 March 2013

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This paper studies low-regularity periodic solutions of the modified two-component Camassa-Holm equation with initial value. We prove the existence and C0-well posedness of solutions.
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02.30.Jr Partial differential equations
02.60.Lj Ordinary and partial differential equations; boundary value problems

Note on intrinsic decay rates for abstract wave equations with memory

Irena Lasiecka, Salim A. Messaoudi, and Muhammad I. Mustafa

J. Math. Phys. 54, 031504 (2013); http://dx.doi.org/10.1063/1.4793988 (18 pages)

Online Publication Date: 13 March 2013

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In this paper we consider a viscoelastic abstract wave equation with memory kernel satisfying the inequality g′ + H(g) ⩽ 0, s ⩾ 0 where H(s) is a given continuous, positive, increasing, and convex function such that H(0) = 0. We shall develop an intrinsic method, based on the main idea introduced by Lasiecka and Tataru [“Uniform boundary stabilization of semilinear wave equation with nonlinear boundary dissipation,” Differential and Integral Equations 6, 507–533 (1993)], for determining decay rates of the energy given in terms of the function H(s). This will be accomplished by expressing the decay rates as a solution to a given nonlinear dissipative ODE. We shall show that the obtained result, while generalizing previous results obtained in the literature, is also capable of proving optimal decay rates for polynomially decaying memory kernels (H(s) ∼ sp) and for the full range of admissible parameters p ∈ [1, 2). While such result has been known for certain restrictive ranges of the parameters p ∈ [1, 3/2), the methods introduced previously break down when p ⩾ 3/2. The present paper develops a new and general tool that is applicable to all admissible parameters.
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02.30.Hq Ordinary differential equations
02.10.De Algebraic structures and number theory

Fractional wave equation and damped waves

Yuri Luchko

J. Math. Phys. 54, 031505 (2013); http://dx.doi.org/10.1063/1.4794076 (16 pages)

Online Publication Date: 13 March 2013

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In this paper, a fractional generalization of the wave equation that describes propagation of damped waves is considered. In contrast to the fractional diffusion-wave equation, the fractional wave equation contains fractional derivatives of the same order α, 1 ⩽ α ⩽ 2, both in space and in time. We show that this feature is a decisive factor for inheriting some crucial characteristics of the wave equation like a constant propagation velocity of both the maximum of its fundamental solution and its gravity and “mass” centers. Moreover, the first, the second, and the Smith centrovelocities of the damped waves described by the fractional wave equation are constant and depend just on the equation order α. The fundamental solution of the fractional wave equation is determined and shown to be a spatial probability density function evolving in time all whose moments of order less than α are finite. To illustrate analytical findings, results of numerical calculations and plots are presented.
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05.60.-k Transport processes
02.50.Cw Probability theory
02.60.Lj Ordinary and partial differential equations; boundary value problems
03.65.Ge Solutions of wave equations: bound states

Global existence and non-relativistic global limits of entropy solutions to the 1D piston problem for the isentropic relativistic Euler equations

Min Ding and Yachun Li

J. Math. Phys. 54, 031506 (2013); http://dx.doi.org/10.1063/1.4792474 (28 pages)

Online Publication Date: 14 March 2013

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We study the 1D piston problem for the isentropic relativistic Euler equations when the total variations of the initial data and the speed of the piston are sufficiently small. Employing a modified Glimm scheme, we establish the global existence of shock front solutions including a strong shock without restriction on the strength. In particular, we give some uniform estimates on the perturbation waves, the reflections of the perturbation waves on the piston and the strong shock. Meanwhile, we consider the convergence of the entropy solutions as the light speed c → +∞ to the corresponding entropy solutions of the classical non-relativistic isentropic Euler equations.
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47.75.+f Relativistic fluid dynamics
05.70.Ce Thermodynamic functions and equations of state
47.40.Nm Shock wave interactions and shock effects

Expansion of the energy of the ground state of the Gross–Pitaevskii equation in the Thomas–Fermi limit

Clément Gallo

J. Math. Phys. 54, 031507 (2013); http://dx.doi.org/10.1063/1.4795245 (13 pages)

Online Publication Date: 18 March 2013

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From the asymptotic expansion of the ground state of the Gross–Pitaevskii equation in the Thomas–Fermi limit given by Gallo and Pelinovsky [“On the Thomas-Fermi ground state in a harmonic potential,” Asymptot. Anal. 73(1–2), 53–96 (2011)]10.3233/ASY-2011-1034, we infer an asymptotic expansion of the kinetic, potential, and total energy of the ground state. In particular, we give a rigorous proof of the expansion of the kinetic energy calculated by Dalfovo, Pitaevskii, and Stringari [“Order parameter at the boundary of a trapped Bose gas,” Phys. Rev. A 54, 4213–4217 (1996)]10.1103/PhysRevA.54.4213 in the case where the space dimension is 3. Moreover, we calculate one more term in this expansion, and we generalize the result to space dimensions 1 and 2.
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03.75.Kk Dynamic properties of condensates; collective and hydrodynamic excitations, superfluid flow
21.60.-n Nuclear structure models and methods

Jet theoretical Yang-Mills energy in the geometric dynamics of two-dimensional monolayer

M. Neagu, N. G. Krylova, and H. V. Grushevskaya

J. Math. Phys. 54, 031508 (2013); http://dx.doi.org/10.1063/1.4795715 (14 pages)

Online Publication Date: 22 March 2013

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Langmuir-Blodgett (LB)-films consist of few LB-monolayers which are high structured nanomaterials that are very promising materials for applications. We use a geometrical approach to describe a structurization into LB-monolayers. Consequently, we develop on the 1-jet space J1([0,∞),math2) the single-time Lagrange geometry (in the sense of distinguished (d-) connection, d-torsions, and an abstract anisotropic electromagnetic-like d-field) for the Lagrangian governing the 2D-motion of a particle of monolayer. One assumed that an expansion near singular points for the constructed geometrical Lagrangian theory describes phase transitions to LB-monolayer. Trajectories of particles in a field of the electrocapillarity forces of monolayer have been calculated in a resonant approximation utilizing a Jacobi equation. A jet geometrical Yang-Mills energy is introduced and some computer graphic simulations are exposed.
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68.47.Pe Langmuir-Blodgett films on solids; polymers on surfaces; biological molecules on surfaces
back to top Quantum Mechanics

The symmetry groups of noncommutative quantum mechanics and coherent state quantization

S. Hasibul Hassan Chowdhury and S. Twareque Ali

J. Math. Phys. 54, 032101 (2013); http://dx.doi.org/10.1063/1.4793992 (21 pages)

Online Publication Date: 8 March 2013

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We explore the group theoretical underpinning of noncommutative quantum mechanics for a system moving on the two-dimensional plane. We show that the pertinent groups for the system are the two-fold central extension of the Galilei group in (2+1)-space-time dimensions and the two-fold extension of the group of translations of math4. This latter group is just the standard Weyl-Heisenberg group of standard quantum mechanics with an additional central extension. We also look at a further extension of this group and discuss its significance to noncommutative quantum mechanics. We build unitary irreducible representations of these various groups and construct the associated families of coherent states. A coherent state quantization of the underlying phase space is then carried out, which is shown to lead to exactly the same commutation relations as usually postulated for this model of noncommutative quantum mechanics.
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03.65.Ta Foundations of quantum mechanics; measurement theory
02.20.-a Group theory
03.65.Fd Algebraic methods

On Pauli pairs

Stanislav Shkarin

J. Math. Phys. 54, 032102 (2013); http://dx.doi.org/10.1063/1.4794087 (10 pages)

Online Publication Date: 11 March 2013

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The state of a system in classical mechanics can be uniquely reconstructed if we know the positions and the momenta of all its parts. In 1958 Pauli has conjectured that the same holds for quantum mechanical systems. The conjecture turned out to be wrong. In this paper we provide a new set of examples of Pauli pairs, being the pairs of quantum states indistinguishable by measuring the spatial location and momentum. In particular, we construct a new set of spatially localized Pauli pairs.
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03.65.Ta Foundations of quantum mechanics; measurement theory

About a new family of coherent states for some SU(1,1) central field potentials

Dusan Popov, Vjekoslav Sajfert, Nicolina Pop, and Viorel Chiritoiu

J. Math. Phys. 54, 032103 (2013); http://dx.doi.org/10.1063/1.4795137 (21 pages)

Online Publication Date: 19 March 2013

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In this paper, we shall define a new family of coherent states which we shall call the “mother coherent states,” bearing in mind the fact that these states are independent from any parameter (the Bargmann index, the rotational quantum number J, and so on). So, these coherent states are defined on the whole Hilbert space of the Fock basis vectors. The defined coherent states are of the Barut-Girardello kind, i.e., they are the eigenstates of the lowering operator. For these coherent states we shall calculate the expectation values of different quantum observables, the corresponding Mandel parameter, the Husimi's distribution function and also the P- function. Finally, we shall particularize the obtained results for the three-dimensional harmonic and pseudoharmonic oscillators.
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03.65.Ge Solutions of wave equations: bound states
02.10.Ud Linear algebra
03.65.Fd Algebraic methods

Quantum graph as a quantum spectral filter

Ondřej Turek and Taksu Cheon

J. Math. Phys. 54, 032104 (2013); http://dx.doi.org/10.1063/1.4795404 (17 pages)

Online Publication Date: 20 March 2013

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We study the transmission of a quantum particle along a straight input–output line to which a graph Γ is attached at a point. In the point of contact we impose a singularity represented by a certain properly chosen scale-invariant coupling with a coupling parameter α. We show that the probability of transmission along the line as a function of the particle energy tends to the indicator function of the energy spectrum of Γ as α → ∞. This effect can be used for a spectral analysis of the given graph Γ. Its applications include a control of a transmission along the line and spectral filtering. The result is illustrated with an example where Γ is a loop exposed to a magnetic field. Two more quantum devices are designed using other special scale-invariant vertex couplings. They can serve as a band-stop filter and as a spectral separator, respectively.
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03.65.Ta Foundations of quantum mechanics; measurement theory
02.10.Ox Combinatorics; graph theory
02.30.-f Function theory, analysis
02.50.Cw Probability theory
03.65.Fd Algebraic methods

Optimal volume Wegner estimate for random magnetic Laplacians on math2

David Hasler and Daniel Luckett

J. Math. Phys. 54, 032105 (2013); http://dx.doi.org/10.1063/1.4794082 (7 pages)

Online Publication Date: 21 March 2013

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We consider a two dimensional magnetic Schrödinger operator on a square lattice with a spatially stationary random magnetic field. We prove a Wegner estimate with optimal volume dependence. The Wegner estimate holds around the spectral edges, and it implies Hölder continuity of the integrated density of states in this region. The proof is based on the Wegner estimate obtained in Erdős and Hasler [“Wegner estimate for random magnetic Laplacians on 2Z2,” Ann. Henri Poincaré 12, 1719–1731 (2012)]10.1007/s00023-012-0177-9.
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03.65.Ge Solutions of wave equations: bound states
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
02.50.-r Probability theory, stochastic processes, and statistics

Group action in topos quantum physics

C. Flori

J. Math. Phys. 54, 032106 (2013); http://dx.doi.org/10.1063/1.4795803 (52 pages)

Online Publication Date: 29 March 2013

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Topos theory has been suggested first by Isham and Butterfield, and then by Isham and Döring, as an alternative mathematical structure within which to formulate physical theories. In particular, it has been used to reformulate standard quantum mechanics in such a way that a novel type of logic is used to represent propositions. In this paper, we extend this formulation to include the notion of a group and group transformation in such a way that we overcome the problem of twisted presheaves. In order to implement this we need to change the type of topos involved, so as to render the notion of continuity of the group action meaningful.
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03.65.Fd Algebraic methods
02.20.Uw Quantum groups
back to top Quantum Information and Computation

A new method to construct families of complex Hadamard matrices in even dimensions

D. Goyeneche

J. Math. Phys. 54, 032201 (2013); http://dx.doi.org/10.1063/1.4794068 (18 pages)

Online Publication Date: 13 March 2013

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We present a new method for constructing affine families of complex Hadamard matrices in every even dimension. This method has an intersection with Diţă’s construction and generalizes Szöllősi's method. We extend some known families and present new ones existing in even dimensions. In particular, we find more than 13 millon inequivalent affine families in dimension 32. We also find analytical restrictions for any set of four mutually unbiased bases existing in dimension six and for any family of complex Hadamard matrices existing in every odd dimension.
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02.10.Yn Matrix theory

Tsirelson's problem and asymptotically commuting unitary matrices

Narutaka Ozawa

J. Math. Phys. 54, 032202 (2013); http://dx.doi.org/10.1063/1.4795391 (8 pages)

Online Publication Date: 15 March 2013

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In this paper, we consider quantum correlations of bipartite systems having a slight interaction, and reinterpret Tsirelson's problem (and hence Kirchberg's and Connes's conjectures) in terms of finite-dimensional asymptotically commuting positive operator valued measures. We also consider the systems of asymptotically commuting unitary matrices and formulate the Stronger Kirchberg Conjecture.
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02.10.Yn Matrix theory
03.65.Fd Algebraic methods

Perturbation bounds for quantum Markov processes and their fixed points

Oleg Szehr and Michael M. Wolf

J. Math. Phys. 54, 032203 (2013); http://dx.doi.org/10.1063/1.4795112 (10 pages)

Online Publication Date: 19 March 2013

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We investigate the stability of quantum Markov processes with respect to perturbations of their transition maps. In the first part, we introduce a condition number that measures the sensitivity of fixed points of a quantum channel to perturbations. We establish upper and lower bounds on this condition number in terms of subdominant eigenvalues of the transition map. In the second part, we consider quantum Markov processes that converge to a unique stationary state and we analyze the stability of the evolution at finite times. In this way we obtain a linear relation between the mixing time of a quantum Markov process and the sensitivity of its fixed point with respect to perturbations of the transition map.
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03.67.Hk Quantum communication
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.10.Ud Linear algebra
02.50.Ga Markov processes

Universal quantum state merging

I. Bjelaković, H. Boche, and G. Janßen

J. Math. Phys. 54, 032204 (2013); http://dx.doi.org/10.1063/1.4795243 (32 pages)

Online Publication Date: 19 March 2013

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We determine the optimal entanglement rate of quantum state merging when assuming that the state is unknown except for its membership in a certain set of states. We find that merging is possible at the lowest rate allowed by the individual states. Additionally, we establish a lower bound for the classical cost of state merging under state uncertainty. To this end we give an elementary proof for the cost in case of a perfectly known state which makes no use of the “resource framework.” As applications of our main result, we determine the capacity for one-way entanglement distillation if the source is not perfectly known. Moreover, we give another achievability proof for the entanglement generation capacity over compound quantum channels.
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03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)
03.67.Hk Quantum communication
03.67.Mn Entanglement measures, witnesses, and other characterizations
back to top Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory

Representing the vacuum polarization on de Sitter

Katie E. Leonard, Tomislav Prokopec, and Richard P. Woodard

J. Math. Phys. 54, 032301 (2013); http://dx.doi.org/10.1063/1.4793987 (9 pages)

Online Publication Date: 6 March 2013

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Previous studies of the vacuum polarization on de Sitter have demonstrated that there is a simple, noncovariant representation of it in which the physics is transparent. There is also a cumbersome, covariant representation in which the physics is obscure. Despite being unwieldy, the latter form has a powerful appeal for those who are concerned about de Sitter invariance. We show that nothing is lost by employing the simple, noncovariant representation because there is a closed form procedure for converting its structure functions to those of the covariant representation. We also present a vastly improved technique for reading off the noncovariant structure functions from the primitive diagrams. And we discuss the issue of representing the vacuum polarization for a general metric background.
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12.20.Ds Specific calculations
98.80.Cq Particle-theory and field-theory models of the early Universe (including cosmic pancakes, cosmic strings, chaotic phenomena, inflationary universe, etc.)

Representations of some quantum tori Lie subalgebras

Jingjing Jiang and Song Wang

J. Math. Phys. 54, 032302 (2013); http://dx.doi.org/10.1063/1.4795350 (15 pages)

Online Publication Date: 20 March 2013

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In this paper, we define the q-analog Virasoro-like Lie subalgebras in x∞ = a∞(b∞, c∞, d∞). The embedding formulas into x∞ are introduced. Irreducible highest weight representations of mathq, mathq, and mathq-series of the q-analog Virasoro-like Lie algebras in terms of vertex operators are constructed. We also construct the polynomial representations of the mathq, mathq, mathq, and mathq-series of the q-analog Virasoro-like Lie algebras.
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02.20.Sv Lie algebras of Lie groups
03.65.Fd Algebraic methods

Towards an invariant geometry of double field theory

Olaf Hohm and Barton Zwiebach

J. Math. Phys. 54, 032303 (2013); http://dx.doi.org/10.1063/1.4795513 (35 pages)

Online Publication Date: 26 March 2013

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We introduce a geometrical framework for double field theory in which generalized Riemann and torsion tensors are defined without reference to a particular basis. This invariant geometry provides a unifying framework for the frame-like and metric-like formulations developed before. We discuss the relation to generalized geometry and give an “index-free” proof of the algebraic Bianchi identity. Finally, we analyze to what extent the generalized Riemann tensor encodes the curvatures of Riemannian geometry. We show that it contains the conventional Ricci tensor and scalar curvature but not the full Riemann tensor, suggesting the possibility of a further extension of this framework.
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02.40.-k Geometry, differential geometry, and topology
45.10.Na Geometrical and tensorial methods
back to top General Relativity and Gravitation

Killing-Yano tensors in spaces admitting a hypersurface orthogonal Killing vector

David Garfinkle and E. N. Glass

J. Math. Phys. 54, 032501 (2013); http://dx.doi.org/10.1063/1.4795122 (10 pages)

Online Publication Date: 21 March 2013

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Methods are presented for finding Killing-Yano tensors, conformal Killing-Yano tensors, and conformal Killing vectors in spacetimes with a hypersurface orthogonal Killing vector. These methods are similar to a method developed by the authors for finding Killing tensors. In all cases one decomposes both the tensor and the equation it satisfies into pieces along the Killing vector and pieces orthogonal to the Killing vector. Solving the separate equations that result from this decomposition requires less computing than integrating the original equation. In each case, examples are given to illustrate the method.
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02.10.Ud Linear algebra
back to top Dynamical Systems

Variational principles for relative local pressure with subadditive potentials

Xianfeng Ma and Ercai Chen

J. Math. Phys. 54, 032701 (2013); http://dx.doi.org/10.1063/1.4794086 (25 pages)

Online Publication Date: 11 March 2013

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We prove two relative local variational principles of topological pressure functions P(T,F,U,y) and P(T,F,U|Y) for a given factor map π: (X, T) → (Y, S) between two topological dynamical systems, an open cover U of X and a subadditive potential F in C(X,math). By proving the upper semi-continuity and affinity of the entropy maps μ→hμ(T,U∣Y) and μ→hμ+(T,U∣Y) on the space of all invariant Borel probability measures, we show that the relative local topological pressure with subadditive potentials determines the local measure-theoretic conditional entropies.
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02.30.Xx Calculus of variations
02.40.Re Algebraic topology
02.50.Cw Probability theory
back to top Classical Mechanics and Classical Fields

Rigid motions: Action-angles, relative cohomology and polynomials with roots on the unit circle

J.-P. Françoise, P. L. Garrido, and G. Gallavotti

J. Math. Phys. 54, 032901 (2013); http://dx.doi.org/10.1063/1.4794089 (19 pages)

Online Publication Date: 13 March 2013

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Revisiting canonical integration of the classical solid near a hyperbolic or elliptic uniform rotation, normal canonical coordinates p, q are constructed so that the Hamiltonian becomes a function (“normal form”) of x+ = pq or of x− = p2 + q2: the two cases are treated simultaneously distinguishing them, respectively, by a label a = ±, in terms of various power series with coefficients which are shown to be polynomials in a variable ra2 depending on the inertia moments. The normal forms are derived via the analysis of a relative cohomology problem and shown to be obtainable without reference to the construction of the normal coordinates via elliptic integrals (unlike the derivation of the normal coordinates p, q). Results and conjectures also emerge about the properties of the above polynomials and the location of their roots. In particular a class of polynomials with all roots on the unit circle arises.
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02.10.De Algebraic structures and number theory
02.30.-f Function theory, analysis

On the Hamilton-Jacobi theory for singular lagrangian systems

Manuel de León, Juan Carlos Marrero, David Martín de Diego, and Miguel Vaquero

J. Math. Phys. 54, 032902 (2013); http://dx.doi.org/10.1063/1.4796088 (32 pages)

Online Publication Date: 26 March 2013

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We develop a Hamilton-Jacobi theory for singular lagrangian systems using the Gotay-Nester-Hinds constraint algorithm. The procedure works even if the system has secondary constraints.
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45.05.+x General theory of classical mechanics of discrete systems
02.30.Jr Partial differential equations
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