Volume 55, Issue 12, December 2014
Index of content:
 ARTICLES

 Partial Differential Equations

The existence of positive solutions with prescribed L ^{2}norm for nonlinear Choquard equations
View Description Hide DescriptionIn this paper, we study the existence of positive solutions with prescribed L ^{2}norm to a class of nonlinear Choquard equation −Δu − λu = (I α ∗F(u)) F′(u) in ℝ^{ N }, where λ ∈ ℝ, N ≥ 3, α ∈ (0, N), I α :ℝ^{ N } → ℝ is the Riesz potential. Under some conditions imposed on F, by using a minimax procedure and the concentration compactness of P. L. Lions, we show that for any c > 0, the equation possesses at least a couple (uc , λc ) ∈ H ^{1}(ℝ^{ N }) × ℝ^{−} of weak solution such that .

Opening up and control of spectral gaps of the Laplacian in periodic domains
View Description Hide DescriptionThe main result of this work is as follows: for arbitrary pairwise disjoint, finite intervals (αj , βj ) ⊂ [0, ∞), j = 1, …, m, and for arbitrary n ≥ 2, we construct a family of periodic noncompact domains {Ω^{ε}⊂ℝ^{ n }}ε>0 such that the spectrum of the Neumann Laplacian in Ω^{ε} has at least m gaps when ε is small enough, moreover the first m gaps tend to the intervals (αj , βj ) as ε → 0. The constructed domain Ω^{ε} is obtained by removing from ℝ^{ n } a system of periodically distributed “traplike” surfaces. The parameter ε characterizes the period of the domain Ω^{ε}, also it is involved in a geometry of the removed surfaces.

Bounded Sobolev norms for KleinGordon equations under nonresonant perturbation
View Description Hide DescriptionIn this paper, we prove Anderson localization for the KleinGordon operator on the circle 𝕋 under nonresonant perturbations. Furthermore, using the result, we show that the Sobolev norms of solutions to the corresponding KleinGordon equations remain bounded for all time.
 ManyBody and Condensed Matter Physics

Quantization of interface currents
View Description Hide DescriptionAt the interface of two twodimensional quantum systems, there may exist interface currents similar to edge currents in quantum Hall systems. It is proved that these interface currents are macroscopically quantized by an integer that is given by the difference of the Chern numbers of the two systems. It is also argued that at the interface between two timereversal invariant systems with halfinteger spin, one of which is trivial and the other nontrivial, there are dissipationless spinpolarized interface currents.
 Quantum Mechanics

Curvatureinduced bound states in Robin waveguides and their asymptotical properties
View Description Hide DescriptionWe analyze bound states of Robin Laplacian in infinite planar domains with a smooth boundary, in particular, their relations to the geometry of the latter. The domains considered have locally straight boundary being, for instance, locally deformed halfplanes or wedges, or infinite strips, alternatively they are the exterior of a bounded obstacle. In the situation when the Robin condition is strongly attractive, we derive a twoterm asymptotic formula in which the nexttoleading term is determined by the extremum of the boundary curvature. We also discuss the nonasymptotic case of attractive boundary interaction and show that the discrete spectrum is nonempty if the domain is a local deformation of a halfplane or a wedge of angle less than π, and it is void if the domain is concave.

Gaussian distributions, Jacobi group, and SiegelJacobi space
View Description Hide DescriptionLet be the space of Gaussian distribution functions over ℝ, regarded as a 2dimensional statistical manifold parameterized by the mean μ and the deviation σ. In this paper, we show that the tangent bundle of , endowed with its natural Kähler structure, is the SiegelJacobi space appearing in the context of Number Theory and Jacobi forms. Geometrical aspects of the SiegelJacobi space are discussed in detail (completeness, curvature, group of holomorphic isometries, space of Kähler functions, and relationship to the Jacobi group), and are related to the quantum formalism in its geometrical form, i.e., based on the Kähler structure of the complex projective space. This paper is a continuation of our previous work [M. Molitor, “Remarks on the statistical origin of the geometrical formulation of quantum mechanics,” Int. J. Geom. Methods Mod. Phys. 9(3), 1220001, 9 (2012); M. Molitor, “Information geometry and the hydrodynamical formulation of quantum mechanics,” eprint arXiv (2012); M. Molitor, “Exponential families, Kähler geometry and quantum mechanics,” J. Geom. Phys. 70, 54–80 (2013)], where we studied the quantum formalism from a geometric and informationtheoretical point of view.

Construction of the BarutGirardello type of coherent states for PöschlTeller potential
View Description Hide DescriptionThe PöschlTeller (PT) potential occupies a privileged place among the anharmonic oscillator potentials due to its applications from quantum mechanics to diatomic molecules. For this potential, a polynomial su(1, 1) algebra has been constructed previously. So far, the coherent states (CSs) associated with this algebra have never appeared. In this paper, we construct the coherent states of the BarutGirardello coherent states (BGCSs) type for the PT potentials, which have received less attention in the scientific literature. We obtain these CSs and demonstrate that they fulfil all conditions required by the coherent state. The Mandel parameter for the pure BGCSs and Husimi’s and Pquasi distributions (for the mixedthermal states) is also presented. Finally, the exponential form of the BGCSs for the PT potential has been presented and enabled us to build Perelomov type CSs for the PT potential. We point out that the BGCSs and the Perelomov type coherent states (PCSs) are related via Laplace transform.

Existence of the StarkWannier quantum resonances
View Description Hide DescriptionIn this paper, we prove the existence of the StarkWannier quantum resonances for onedimensional Schrödinger operators with smooth periodic potential and small external homogeneous electric field. Such a result extends the existence result previously obtained in the case of periodic potentials with a finite number of open gaps.

SU(4) based classification of fourlevel systems and their semiclassical solution
View Description Hide DescriptionWe present a systematic method to classify the fourlevel system using SU (4) symmetry as the basis group. It is shown that this symmetry allows three dipole transitions which eventually leads to six possible configurations of the fourlevel system. Using a dressed atom approach, the semiclassical version of each configuration is exactly solved under rotating wave approximation and the symmetry among the Rabi oscillation among various models is studied.
 Quantum Information and Computation

Positivity, discontinuity, finite resources, and nonzero error for arbitrarily varying quantum channels
View Description Hide DescriptionThis work is motivated by a quite general question: Under which circumstances are the capacities of information transmission systems continuous? The research is explicitly carried out on finite arbitrarily varying quantum channels (AVQCs). We give an explicit example that answers the recent question whether the transmission of messages over AVQCs can benefit from assistance by distribution of randomness between the legitimate sender and receiver in the affirmative. The specific class of channels introduced in that example is then extended to show that the unassisted capacity does have discontinuity points, while it is known that the randomnessassisted capacity is always continuous in the channel. We characterize the discontinuity points and prove that the unassisted capacity is always continuous around its positivity points. After having established shared randomness as an important resource, we quantify the interplay between the distribution of finite amounts of randomness between the legitimate sender and receiver, the (nonzero) probability of a decoding error with respect to the average error criterion and the number of messages that can be sent over a finite number of channel uses. We relate our results to the entanglement transmission capacities of finite AVQCs, where the role of shared randomness is not yet well understood, and give a new sufficient criterion for the entanglement transmission capacity with randomness assistance to vanish.

How to efficiently select an arbitrary Clifford group element
View Description Hide DescriptionWe give an algorithm which produces a unique element of the Clifford group on n qubits ( ) from an integer (the number of elements in the group). The algorithm involves O(n ^{3}) operations and provides, in addition to a canonical mapping from the integers to group elements g, a factorization of g into a sequence of at most 4n symplectic transvections. The algorithm can be used to efficiently select random elements of which are often useful in quantum information theory and quantum computation. We also give an algorithm for the inverse map, indexing a group element in time O(n ^{3}).

Entanglement and the threedimensionality of the Bloch ball
View Description Hide DescriptionWe consider a very natural generalization of quantum theory by letting the dimension of the Bloch ball be not necessarily three. We analyze bipartite state spaces where each of the components has a ddimensional Euclidean ball as state space. In addition to this, we impose two very natural assumptions: the continuity and reversibility of dynamics and the possibility of characterizing bipartite states by local measurements. We classify all these bipartite state spaces and prove that, except for the quantum twoqubit state space, none of them contains entangled states. Equivalently, in any of these nonquantum theories, interacting dynamics is impossible. This result reveals that “existence of entanglement” is the requirement with minimal logical content which singles out quantum theory from our family of theories.
 Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory

Nongeometric fluxes, quasiHopf twist deformations, and nonassociative quantum mechanics
View Description Hide DescriptionWe analyse the symmetries underlying nonassociative deformations of geometry in nongeometric Rflux compactifications which arise via Tduality from closed strings with constant geometric fluxes. Starting from the nonabelian Lie algebra of translations and Bopp shifts in phase space, together with a suitable cochain twist, we construct the quasiHopf algebra of symmetries that deforms the algebra of functions and the exterior differential calculus in the phase space description of nonassociative Rspace. In this setting, nonassociativity is characterised by the associator 3cocycle which controls noncoassociativity of the quasiHopf algebra. We use abelian 2cocycle twists to construct maps between the dynamical nonassociative star product and a family of associative star products parametrized by constant momentum surfaces in phase space. We define a suitable integration on these nonassociative spaces and find that the usual cyclicity of associative noncommutative deformations is replaced by weaker notions of 2cyclicity and 3cyclicity. Using this star product quantization on phase space together with 3cyclicity, we formulate a consistent version of nonassociative quantum mechanics, in which we calculate the expectation values of area and volume operators, and find coarsegraining of the string background due to the Rflux.

On the AshtekarLewandowski measure as a restriction of the product one
View Description Hide DescriptionIt is known that the kdimensional Hausdorff measure on a kdimensional submanifold of ℝ^{ n } is closely related to the Lebesgue measure on ℝ^{ n }. We show that the AshtekarLewandowski measure on the space of generalized Gconnections for a compact, connected, semisimple Lie group G is analogously related to the product measure on the set of all Gvalued functions on the group of loops. We also show that, under very mild conditions, the AshtekarLewandowski measure is supported on nowherecontinuous generalized connections.

Dimensional regularization in position space and a Forest Formula for EpsteinGlaser renormalization
View Description Hide DescriptionWe reformulate dimensional regularization as a regularization method in position space and show that it can be used to give a closed expression for the renormalized timeordered products as solutions to the induction scheme of EpsteinGlaser. This closed expression, which we call the EpsteinGlaser Forest Formula, is analogous to Zimmermann’s Forest Formula for BPH renormalization. For scalar fields, the resulting renormalization method is always applicable, we compute several examples. We also analyze the Hopf algebraic aspects of the combinatorics. Our starting point is the Main Theorem of Renormalization of Stora and Popineau and the arising renormalization group as originally defined by Stückelberg and Petermann.

Dipoles in graphene have infinitely many bound states
View Description Hide DescriptionWe show that in graphene, modelled by the twodimensional Dirac operator, charge distributions with nonvanishing dipole moment have infinitely many bound states. The corresponding eigenvalues accumulate at the edges of the gap faster than any power.
 Dynamical Systems

The TremblayTurbinerWinternitz system as extended Hamiltonian
View Description Hide DescriptionWe generalize the idea of “extension of Hamiltonian systems”—developed in a series of previous articles—which allows the explicit construction of Hamiltonian systems with additional nontrivial polynomial first integrals of arbitrarily high degree, as well as the determination of new superintegrable systems from old ones. The present generalization, that we call “modified extension of Hamiltonian systems,” produces the third independent first integral for the (complete) TremblayTurbinerWinternitz system, as well as for the caged anisotropic oscillator in dimension two.

Conformal killing tensors and covariant Hamiltonian dynamics
View Description Hide DescriptionA covariant algorithm for deriving the conserved quantities for natural Hamiltonian systems is combined with the nonrelativistic framework of Eisenhart, and of Duval, in which the classical trajectories arise as geodesics in a higher dimensional spacetime, realized by Brinkmann manifolds. Conserved quantities which are polynomial in the momenta can be built using timedependent conformal Killing tensors with flux. The latter are associated with terms proportional to the Hamiltonian in the lower dimensional theory and with spectrum generating algebras for higher dimensional quantities of order 1 and 2 in the momenta. Illustrations of the general theory include the RungeLenz vector for planetary motion with a timedependent gravitational constant G(t), motion in a timedependent electromagnetic field of a certain form, quantum dots, the HénonHeiles and Holt systems, respectively, providing us with Killing tensors of rank that ranges from one to six.
 Classical Mechanics and Classical Fields

Lagrangian approach to the physical degree of freedom count
View Description Hide DescriptionIn this paper, we present a Lagrangian method that allows the physical degree of freedom count for any Lagrangian system without having to perform neither Dirac nor covariant canonical analyses. The essence of our method is to establish a map between the relevant Lagrangian parameters of the current approach and the Hamiltonian parameters that enter in the formula for the counting of the physical degrees of freedom as is given in Dirac’s method. Once the map is obtained, the usual Hamiltonian formula for the counting can be expressed in terms of Lagrangian parameters only, and therefore we can remain in the Lagrangian side without having to go to the Hamiltonian one. Using the map, it is also possible to count the number of first and secondclass constraints within the Lagrangian formalism only. For the sake of completeness, the geometric structure underlying the current approach—developed for systems with a finite number of degrees of freedom—is uncovered with the help of the covariant canonical formalism. Finally, the method is illustrated in several examples, including the relativistic free particle.

The action principle for dissipative systems
View Description Hide DescriptionIn the present work, we redefine and generalize the action principle for dissipative systems proposed by Riewe by fixing the mathematical inconsistencies present in the original approach. In order to formulate a quadratic Lagrangian for nonconservative systems, the Lagrangian functions proposed depend on mixed integer order and fractional order derivatives. As examples, we formulate a quadratic Lagrangian for a particle under a frictional force proportional to the velocity and to the classical problem of an accelerated point charge.