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Volume 54, Issue 3, March 2013

We consider a protoring nebula of a gas giant such as Neptune as a cloud of gas/dust particles whose behaviour is governed by the stochastic mechanics associated to the Kepler problem. This leads to a stochastic BurgersZeldovich type model for the formation of planetesimals involving a stochastic Burgers equation with vorticity which could help to explain the turbulent behaviour observed in ring systems. The Burgers fluid density and the distribution of the mass M(T) of a spherical planetesimal of radius δ are computed for times T. For circular orbits, sufficient conditions on certain time averages of δ^{2} are given ensuring that VarM(T) ∼ 0 as T ∼ ∞. Some applications are given to the satellites of Jupiter and Saturn, in particular giving a possible explanation of the equal mass families of satellites.
 ARTICLES

 Partial Differential Equations

Ground state solutions for nonlinear fractional Schrödinger equations in
View Description Hide DescriptionWe 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.

Newtonian limit and trend to equilibrium for the relativistic FokkerPlanck equation
View Description Hide DescriptionThe relativistic FokkerPlanck 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 L ^{1} to solutions of the nonrelativistic FokkerPlanck 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 L ^{1} 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 L ^{2} norm.

Lowregularity solutions of the periodic modified twocomponent CamassaHolm equation
View Description Hide DescriptionThis paper studies lowregularity periodic solutions of the modified twocomponent CamassaHolm equation with initial value. We prove the existence and C ^{0}well posedness of solutions.

Note on intrinsic decay rates for abstract wave equations with memory
View Description Hide DescriptionIn 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 Equations6, 507–533 (Year: 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) ∼ s ^{ p }) 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.

Fractional wave equation and damped waves
View Description Hide DescriptionIn this paper, a fractional generalization of the wave equation that describes propagation of damped waves is considered. In contrast to the fractional diffusionwave 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.

Global existence and nonrelativistic global limits of entropy solutions to the 1D piston problem for the isentropic relativistic Euler equations
View Description Hide DescriptionWe 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 nonrelativistic isentropic Euler equations.

Expansion of the energy of the ground state of the Gross–Pitaevskii equation in the Thomas–Fermi limit
View Description Hide DescriptionFrom the asymptotic expansion of the ground state of the Gross–Pitaevskii equation in the Thomas–Fermi limit given by Gallo and Pelinovsky [“On the ThomasFermi ground state in a harmonic potential,” Asymptot. Anal.73(1–2), 53–96 (Year: 2011)]10.3233/ASY20111034, 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. A54, 4213–4217 (Year: 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.

Jet theoretical YangMills energy in the geometric dynamics of twodimensional monolayer^{a)}
View Description Hide DescriptionLangmuirBlodgett (LB)films consist of few LBmonolayers which are high structured nanomaterials that are very promising materials for applications. We use a geometrical approach to describe a structurization into LBmonolayers. Consequently, we develop on the 1jet space the singletime Lagrange geometry (in the sense of distinguished (d) connection, dtorsions, and an abstract anisotropic electromagneticlike dfield) for the Lagrangian governing the 2Dmotion of a particle of monolayer. One assumed that an expansion near singular points for the constructed geometrical Lagrangian theory describes phase transitions to LBmonolayer. 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 YangMills energy is introduced and some computer graphic simulations are exposed.
 Quantum Mechanics

The symmetry groups of noncommutative quantum mechanics and coherent state quantization
View Description Hide DescriptionWe explore the group theoretical underpinning of noncommutative quantum mechanics for a system moving on the twodimensional plane. We show that the pertinent groups for the system are the twofold central extension of the Galilei group in (2+1)spacetime dimensions and the twofold extension of the group of translations of . This latter group is just the standard WeylHeisenberg 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.

On Pauli pairs
View Description Hide DescriptionThe 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.

About a new family of coherent states for some SU(1,1) central field potentials
View Description Hide DescriptionIn 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 BarutGirardello 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 threedimensional harmonic and pseudoharmonic oscillators.

Quantum graph as a quantum spectral filter
View Description Hide DescriptionWe 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 scaleinvariant 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 scaleinvariant vertex couplings. They can serve as a bandstop filter and as a spectral separator, respectively.

Optimal volume Wegner estimate for random magnetic Laplacians on
View Description Hide DescriptionWe 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 ,” Ann. Henri Poincaré12, 1719–1731 (Year: 2012)]10.1007/s0002301201779.

Group action in topos quantum physics
View Description Hide DescriptionTopos 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.
 Quantum Information and Computation

A new method to construct families of complex Hadamard matrices in even dimensions
View Description Hide DescriptionWe 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.

Tsirelson's problem and asymptotically commuting unitary matrices
View Description Hide DescriptionIn 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 finitedimensional asymptotically commuting positive operator valued measures. We also consider the systems of asymptotically commuting unitary matrices and formulate the Stronger Kirchberg Conjecture.

Perturbation bounds for quantum Markov processes and their fixed points
View Description Hide DescriptionWe 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.

Universal quantum state merging
View Description Hide DescriptionWe 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 oneway entanglement distillation if the source is not perfectly known. Moreover, we give another achievability proof for the entanglement generation capacity over compound quantum channels.
 Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory

Representing the vacuum polarization on de Sitter
View Description Hide DescriptionPrevious 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.

Representations of some quantum tori Lie subalgebras
View Description Hide DescriptionIn this paper, we define the qanalog Virasorolike Lie subalgebras in x _{∞} = a _{∞}(b _{∞}, c _{∞}, d _{∞}). The embedding formulas into x _{∞} are introduced. Irreducible highest weight representations of , , and series of the qanalog Virasorolike Lie algebras in terms of vertex operators are constructed. We also construct the polynomial representations of the , , , and series of the qanalog Virasorolike Lie algebras.