Volume 54, Issue 11, November 2013
Index of content:

We investigate the role that vortex loops play in characterizing eigenstates of interacting Majoranas. We give some general results and then focus on ladder Hamiltonian examples as a test of further ideas. Two methods yield exact results: (i) A mapping of certain spin Hamiltonians to quartic interactions of Majoranas shows that the spectra of these two examples coincide. (ii) In cases with reflectionsymmetric Hamiltonians, we use reflection positivity for Majoranas to characterize vortices in the ground states. Two additional methods suggest wider applicability of these results: (iii) Numerical evidence suggests similar behavior for certain systems without reflection symmetry. (iv) A perturbative analysis also suggests similar behavior without the assumption of reflection symmetry.
 ARTICLES


Partial Differential Equations

An extended magnetostatic BornInfeld model with a concave lower order term
View Description Hide DescriptionThis paper concerns an extended BornInfeld model with a concave lower order term for the magnetostatic case. Three types of boundary value problems are considered: the boundary condition prescribing the tangential component of A, the natural boundary condition, and the boundary condition prescribing the tangential component of . In each case we obtain existence and regularity of solutions for small boundary data.

Presymplectic current and the inverse problem of the calculus of variations
View Description Hide DescriptionThe inverse problem of the calculus of variations asks whether a given system of partial differential equations (PDEs) admits a variational formulation. We show that the existence of a presymplectic form in the variational bicomplex, when horizontally closed on solutions, allows us to construct a variational formulation for a subsystem of the given PDE. No constraints on the differential order or number of dependent or independent variables are assumed. The proof follows a recent observation of Bridges, Hydon, and Lawson [Math. Proc. Cambridge Philos. Soc. 148(01), 159–178 (2010)] and generalizes an older result of Henneaux [Ann. Phys. 140(1), 45–64 (1982)] from ordinary differential equations (ODEs) to PDEs. Uniqueness of the variational formulation is also discussed.

Spectral asymmetry of the massless Dirac operator on a 3torus
View Description Hide DescriptionConsider the massless Dirac operator on a 3torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. The aim of the paper is to develop a perturbation theory for the eigenvalue with smallest modulus with respect to perturbations of the metric. Here the application of perturbation techniques is hindered by the fact that eigenvalues of the massless Dirac operator have even multiplicity, which is a consequence of this operator commuting with the antilinear operator of charge conjugation (a peculiar feature of dimension 3). We derive an asymptotic formula for the eigenvalue with smallest modulus for arbitrary perturbations of the metric and present two particular families of Riemannian metrics for which the eigenvalue with smallest modulus can be evaluated explicitly. We also establish a relation between our asymptotic formula and the eta invariant.

Asymptotics for inhomogeneous Dirichlet initialboundary value problem for the nonlinear Schrödinger equation
View Description Hide DescriptionWe consider the inhomogeneous Dirichlet initialboundary value problem for the nonlinear Schrödinger equation, formulated on a halfline. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initialboundary value problem and the asymptotic behavior of solutions for large time.

Standing waves for coupled nonlinear Schrödinger equations with decaying potentials
View Description Hide DescriptionWe study the following singularly perturbed problem for a coupled nonlinear Schrödinger system which arises in BoseEinstein condensate: −ε^{2}Δu + a(x)u = μ1 u ^{3} + βuv ^{2} and −ε^{2}Δv + b(x)v = μ2 v ^{3} + βu ^{2} v in with u, v > 0 and u(x), v(x) → 0 as x → ∞. Here, a, b are nonnegative continuous potentials, and μ1, μ2 > 0. We consider the case where the coupling constant β > 0 is relatively large. Then for sufficiently small ɛ > 0, we obtain positive solutions of this system which concentrate around local minima of the potentials as ɛ → 0. The novelty is that the potentials a and b may vanish at someplace and decay to 0 at infinity.

ManyBody and Condensed Matter Physics

Multitransmissionlinebeam interactive system
View Description Hide DescriptionWe construct here a Lagrangian field formulation for a system consisting of an electron beam interacting with a slowwave structure modeled by a possibly nonuniform multiple transmission line (MTL). In the case of a single line we recover the linear model of a traveling wave tube due to J. R. Pierce. Since a properly chosen MTL can approximate a real waveguide structure with any desired accuracy, the proposed model can be used in particular for design optimization. Furthermore, the Lagrangian formulation provides: (i) a clear identification of the mathematical source of amplification, (ii) exact expressions for the conserved energy and its flux distributions obtained from the Noether theorem. In the case of uniform MTLs we carry out an exhaustive analysis of eigenmodes and find sharp conditions on the parameters of the system to provide for amplifying regimes.

Quantum Mechanics

On a Riesz basis of exponentials related to the eigenvalues of an analytic operator and application to a nonselfadjoint problem deduced from a perturbation method for sound radiation
View Description Hide DescriptionIn the present paper, we prove that the family of exponentials associated to the eigenvalues of the perturbed operator T(ɛ) ≔ T 0 + ɛT 1 + ɛ^{2} T 2 + … + ɛ^{ k } T k + … forms a Riesz basis in L ^{2}(0, T), T > 0, where , T 0 is a closed densely defined linear operator on a separable Hilbert space with domain having isolated eigenvalues with multiplicity one, while T 1, T 2, … are linear operators on having the same domain and satisfying a specific growing inequality. After that, we generalize this result using a HLipschitz function. As application, we consider a nonselfadjoint problem deduced from a perturbation method for sound radiation.

Emergence of complex and spinor wave functions in scale relativity. I. Nature of scale variables
View Description Hide DescriptionOne of the main results of scale relativity as regards the foundation of quantum mechanics is its explanation of the origin of the complex nature of the wave function. The scale relativity theory introduces an explicit dependence of physical quantities on scale variables, founding itself on the theorem according to which a continuous and nondifferentiable spacetime is fractal (i.e., scaledivergent). In the present paper, the nature of the scale variables and their relations to resolutions and differential elements are specified in the nonrelativistic case (fractal space). We show that, owing to the scaledependence which it induces, nondifferentiability involves a fundamental twovaluedness of the mean derivatives. Since, in the scale relativity framework, the wave function is a manifestation of the velocity field of fractal spacetime geodesics, the twovaluedness of velocities leads to write them in terms of complex numbers, and yields therefore the complex nature of the wave function, from which the usual expression of the Schrödinger equation can be derived.

Combinatorial theory of the semiclassical evaluation of transport moments. I. Equivalence with the random matrix approach
View Description Hide DescriptionTo study electronic transport through chaotic quantum dots, there are two main theoretical approaches. One involves substituting the quantum system with a random scattering matrix and performing appropriate ensemble averaging. The other treats the transport in the semiclassical approximation and studies correlations among sets of classical trajectories. There are established evaluation procedures within the semiclassical evaluation that, for several linear and nonlinear transport moments to which they were applied, have always resulted in the agreement with random matrix predictions. We prove that this agreement is universal: any semiclassical evaluation within the accepted procedures is equivalent to the evaluation within random matrix theory. The equivalence is shown by developing a combinatorial interpretation of the trajectory sets as ribbon graphs (maps) with certain properties and exhibiting systematic cancellations among their contributions. Remaining trajectory sets can be identified with primitive (palindromic) factorisations whose number gives the coefficients in the corresponding expansion of the moments of random matrices. The equivalence is proved for systems with and without time reversal symmetry.

Heat kernel asymptotics for magnetic Schrödinger operators
View Description Hide DescriptionWe explicitly construct parametrices for magnetic Schrödinger operators on and prove that they provide a complete smallt expansion for the corresponding heat kernel, both on and off the diagonal.

Quantum mechanics with coordinate dependent noncommutativity
View Description Hide DescriptionNoncommutative quantum mechanics can be considered as a first step in the construction of quantum field theory on noncommutative spaces of generic form, when the commutator between coordinates is a function of these coordinates. In this paper we discuss the mathematical framework of such a theory. The noncommutativity is treated as an external antisymmetric field satisfying the Jacobi identity. First, we propose a symplectic realization of a given Poisson manifold and construct the Darboux coordinates on the obtained symplectic manifold. Then we define the star product on a Poisson manifold and obtain the expression for the trace functional. The above ingredients are used to formulate a nonrelativistic quantum mechanics on noncommutative spaces of general form. All considered constructions are obtained as a formal series in the parameter of noncommutativity. In particular, the complete algebra of commutation relations between coordinates and conjugated momenta is a deformation of the standard Heisenberg algebra. As examples we consider a free particle and an isotropic harmonic oscillator on the rotational invariant noncommutative space.

Spectral singularity in confined symmetric optical potential
View Description Hide DescriptionWe present an analytical study for the scattering amplitudes (Reflection R and Transmission T), of the periodic symmetric optical potential confined within the region 0 ⩽ x ⩽ L, embedded in a homogeneous medium having uniform potential W 0. The confining length L is considered to be some integral multiple of the period π. We give some new and interesting results. Scattering is observed to be normal (T^{2} ⩽ 1, R^{2} ⩽ 1) for V 0 ⩽ 0.5, when the above potential can be mapped to a Hermitian potential by a similarity transformation. Beyond this point (V 0 > 0.5) scattering is found to be anomalous (T^{2}, R^{2} not necessarily ⩽1). Additionally, in this parameter regime of V 0, one observes infinite number of spectral singularities E SS at different values of V 0. Furthermore, for L = 2nπ, the transition point V 0 = 0.5 shows unidirectional invisibility with zero reflection when the beam is incident from the absorptive side (Im[V(x)] < 0) but with finite reflection when the beam is incident from the emissive side (Im[V(x)] > 0), transmission being identically unity in both cases. Finally, the scattering coefficients R^{2} and T^{2} always obey the generalized unitarity relation : , where subscripts R and L stand for right and left incidence, respectively.

An exactly solvable threedimensional nonlinear quantum oscillator
View Description Hide DescriptionExact analytical, closedform solutions, expressed in terms of special functions, are presented for the case of a threedimensional nonlinear quantum oscillator with a position dependent mass. This system is the generalization of the corresponding onedimensional system, which has been the focus of recent attention. In contrast to other approaches, we are able to obtain solutions in terms of special functions, without a reliance upon a Rodriguestype of formula. The wave functions of the quantum oscillator have the familiar spherical harmonic solutions for the angular part. For the sstates of the system, the radial equation accepts solutions that have been recently found for the onedimensional nonlinear quantum oscillator, given in terms of associated Legendre functions, along with a constant shift in the energy eigenvalues. Radial solutions are obtained for all angular momentum states, along with the complete energy spectrum of the bound states.

Quantum Information and Computation

Reversibility of a quantum channel: General conditions and their applications to Bosonic linear channels
View Description Hide DescriptionThe method of complementary channel for analysis of reversibility (sufficiency) of a quantum channel with respect to families of input states (pure states for the most part) are considered and applied to Bosonic linear (quasifree) channels, in particular, to Bosonic Gaussian channels. The obtained reversibility conditions for Bosonic linear channels have clear physical interpretation and their sufficiency is also shown by explicit construction of reversing channels. The method of complementary channel gives possibility to prove necessity of these conditions and to describe all reversed families of pure states in the Schrodinger representation. Some applications in quantum information theory are considered. Conditions for existence of discrete classicalquantum subchannels and of completely depolarizing subchannels of a Bosonic linear channel are presented.

Arbitrarily small amounts of correlation for arbitrarily varying quantum channels
View Description Hide DescriptionAs our main result show that in order to achieve the randomness assisted message and entanglement transmission capacities of a finite arbitrarily varying quantum channel it is not necessary that sender and receiver share (asymptotically perfect) common randomness. Rather, it is sufficient that they each have access to an unlimited amount of uses of one part of a correlated bipartite source. This access might be restricted to an arbitrary small (nonzero) fraction per channel use, without changing the main result. We investigate the notion of common randomness. It turns out that this is a very costly resource – generically, it cannot be obtained just by local processing of a bipartite source. This result underlines the importance of our main result. Also, the asymptotic equivalence of the maximal and average error criterion for classical message transmission over finite arbitrarily varying quantum channels is proven. At last, we prove a simplified symmetrizability condition for finite arbitrarily varying quantum channels.

Vortex loops and Majoranas
View Description Hide DescriptionWe investigate the role that vortex loops play in characterizing eigenstates of interacting Majoranas. We give some general results and then focus on ladder Hamiltonian examples as a test of further ideas. Two methods yield exact results: (i) A mapping of certain spin Hamiltonians to quartic interactions of Majoranas shows that the spectra of these two examples coincide. (ii) In cases with reflectionsymmetric Hamiltonians, we use reflection positivity for Majoranas to characterize vortices in the ground states. Two additional methods suggest wider applicability of these results: (iii) Numerical evidence suggests similar behavior for certain systems without reflection symmetry. (iv) A perturbative analysis also suggests similar behavior without the assumption of reflection symmetry.

Interpolatability distinguishes LOCC from separable von Neumann measurements
View Description Hide DescriptionLocal operations with classical communication (LOCC) and separable operations are two classes of quantum operations that play key roles in the study of quantum entanglement. Separable operations are strictly more powerful than LOCC, but no simple explanation of this phenomenon is known. We show that, in the case of von Neumann measurements, the ability to interpolate measurements is an operational principle that sets apart LOCC and separable operations.

Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory

On the exact evaluation of spin networks
View Description Hide DescriptionWe introduce a fully coherent spin network amplitude whose expansion generates all SU(2) spin networks associated with a given graph. We then give an explicit evaluation of this amplitude for an arbitrary graph. We show how this coherent amplitude can be obtained from the specialization of a generating functional obtained by the contraction of parametrized intertwiners à la Schwinger. We finally give the explicit evaluation of this generating functional for arbitrary graphs.

A first look at Weyl anomalies in shape dynamics
View Description Hide DescriptionOne of the more popular objections towards shape dynamics is the suspicion that anomalies in the spatial Weyl symmetry will arise upon quantization. The purpose of this short paper is to establish the tools required for an investigation of the sort of anomalies that can possibly arise. The first step is to adapt to our setting Barnich and Henneaux's formulation of gauge cohomology in the Hamiltonian setting, which serve to decompose the anomaly into a spatial component and time component. The spatial part of the anomaly, i.e., the anomaly in the symmetry algebra itself ([Ω, Ω] ∝ ℏ instead of vanishing) is given by a projection of the second ghost cohomology of the Hamiltonian BRST differential associated to Ω, modulo spatial derivatives. The temporal part, [Ω, H] ∝ ℏ is given by a different projection of the first ghost cohomology and an extra piece arising from a solution to a functional differential equation. Assuming locality of the gauge cohomology groups involved, this part is always local. Assuming locality for the gauge cohomology groups, using Barnich and Henneaux's results, the classification of Weyl cohomology for higher ghost numbers performed by Boulanger, and following the descent equations, we find a complete characterizations of anomalies in 3+1 dimensions. The spatial part of the anomaly and the first component of the temporal anomaly are always local given these assumptions even in shape dynamics. The part emerging from the solution of the functional differential equations explicitly involves the shape dynamics Hamiltonian, and thus might be nonlocal. If one restricts this extra piece of the temporal anomaly to be also local, then overall no Weyl anomalies, either temporal or spatial, emerge in the 3+1 case.

Fluids

Wave speeds in the macroscopic extended model for ultrarelativistic gases
View Description Hide DescriptionEquations determining wave speeds for a model of ultrarelativistic gases are investigated. This model is already present in literature; it deals with an arbitrary number of moments and it was proposed in the context of exact macroscopic approaches in Extended Thermodynamics. We find these results: the whole system for the determination of the wave speeds can be divided into independent subsystems which are expressed by linear combinations, through scalar coefficients, of tensors all of the same order; some wave speeds, but not all of them, are expressed by square roots of rational numbers; finally, we prove that these wave speeds for the macroscopic model are the same of those furnished by the kinetic model.
