Volume 54, Issue 6, June 2013
Index of content:

A global analysis is presented of solutions for Laplace's equation on threedimensional Euclidean space in one of the most general orthogonal asymmetric confocal cyclidic coordinate systems which admit solutions through separation of variables. We refer to this coordinate system as fivecyclide coordinates since the coordinate surfaces are given by two cyclides of genus zero which represent inversions of each other with respect to the unit sphere, a cyclide of genus one, and two disconnected cyclides of genus zero. This coordinate system is obtained by stereographic projection of spheroconal coordinates on fourdimensional Euclidean space. The harmonics in this coordinate system are given by products of solutions of secondorder Fuchsian ordinary differential equations with five elementary singularities. The Dirichlet problem for the global harmonics in this coordinate system is solved using multiparameter spectral theory in the regions bounded by the asymmetric confocal cyclidic coordinate surfaces.
 ARTICLES


Partial Differential Equations

Hölder continuity of the solution map for the Novikov equation
View Description Hide DescriptionThe Novikov equation (NE) has been discovered recently as a new integrable equation with cubic nonlinearities that is similar to the CamassaHolm and DegasperisProcesi equations, which have quadratic nonlinearities. NE is wellposed in Sobolev spaces H s on both the line and the circle for s > 3/2, in the sense of Hadamard, and its datatosolution map is continuous but not uniformly continuous. This work studies the continuity properties of NE further. For initial data in H s , s > 3/2, it is shown that the solution map for NE is Hölder continuous in H r topology for all 0 ⩽ r < s with exponent α depending on s and r.

Photonic lattices
View Description Hide DescriptionWe use critical point theory to find periodic solutions of the nonlinear steady state Schrödinger equations arising in the study of photonic lattices. We show that nontrivial solutions exist for wide ranges of the parameters. It follows that there is a large continuous energy or wavenumber spectrum that allows the existence of steady state solutions. Our results hold in arbitrary dimensions.

Orbital stability of a two parameter family of solitary waves for a fourth order nonlinear Schrödinger type equation
View Description Hide DescriptionThis paper is concerned with the orbital stability of a two parameter family of solitary waves for the fourth order nonlinear Schrödinger type equation (4NLS). By using variational approach, we prove that those solitary waves are orbitally stable under the time evolution by 4NLS.

Ground state solutions of asymptotically linear fractional Schrödinger equations
View Description Hide DescriptionThis paper is devoted to a timeindependent fractional Schrödinger equation of the form where N ⩾ 2, s ∈ (0, 1), (−Δ) s stands for the fractional Laplacian. We apply the variational methods to obtain the existence of ground state solutions when f(x, u) is asymptotically linear with respect to u at infinity.

Existence and concentration of semiclassical solutions for a nonlinear MaxwellDirac system
View Description Hide DescriptionWe study semiclassical ground states of nonlinear MaxwellDirac system with critical/subcritical nonlinearities: α · (iℏ∇ + q(x)A(x))w − aβw − ωw − q(x)ϕ(x)w = f(x, w)w, − Δϕ = q(x)w2, and where , A = (A 1, A 2, A 3) is the magnetic field, ϕ is the electron field, and q is the changing pointwise charge distribution. We develop a variational argument to establish the existence of least energy solutions for ℏ small. We also describe the concentration phenomena of the solutions as ℏ → 0.

Quantum Mechanics

Birth and death processes and quantum spin chains
View Description Hide DescriptionThis paper underscores the intimate connection between the quantum walks generated by certain semiinfinite spin chain Hamiltonians and classical birth and death processes. It is observed that transition amplitudes between single excitation states of the spin chains have an expression in terms of orthogonal polynomials which is analogous to the KarlinMcGregor representation formula of the transition probability functions for classes of birth and death processes. As an application, we present a characterization of spin systems for which the probability to return to the point of origin at some time is 1 or almost 1.

Effects of the magnetic field direction and anisotropy on the interband light absorption of an asymmetric quantum dot
View Description Hide DescriptionIn this paper, the direct interband transition and the threshold frequency of absorption in a twodimensional anisotropic quantum dot are studied under the influence of a tilted external magnetic field. We first calculate the analytical wave functions and energy levels using a transformation to simplify the Hamiltonian of the system. Then, we obtain the analytical expressions for the light interband absorption coefficient and the threshold frequency of absorption as a function of the magnetic field, magnetic field direction, and anisotropy of the system. According to the results obtained from the present work, we find that (i) the absorption threshold frequency (ATF) increases when the magnetic field increases for all directions. (ii) When anisotropy is increased, ATF increases. (iii) At small anisotropy, the magnetic field direction has no important effect on the ATF. In brief, the magnetic field, magnetic field direction, and anisotropy play important roles in the ATF and absorption coefficient.

On an extension problem for density matrices
View Description Hide DescriptionWe investigate the problem of the existence of a density matrix ρ123 on a Hilbert space with given partial traces ρ12 = Tr3 ρ123 and ρ23 = Tr1 ρ123. While we do not solve this problem completely, we offer partial results in the form of some necessary and some sufficient conditions on ρ12 and ρ23. The quantum case differs markedly from the classical (commutative) case, where the obvious necessary compatibility condition suffices, namely, Tr1 ρ12 = Tr3ρ23.

Coarsegrained spin densityfunctional theory: Infinitevolume limit via the hyperfinite
View Description Hide DescriptionCoarsegrained spin density functional theory (SDFT) is a version of SDFT which works with number/spin densities specified to a limited resolution — averages over cells of a regular spatial partition — and external potentials constant on the cells. This coarsegrained setting facilitates a rigorous investigation of the mathematical foundations which goes well beyond what is currently possible in the conventional formulation. Problems of existence, uniqueness, and regularity of representing potentials in the coarsegrained SDFT setting are here studied using techniques of (Robinsonian) nonstandard analysis. Every density which is nowhere spinsaturated is Vrepresentable, and the set of representing potentials is the functional derivative, in an appropriate generalized sense, of the Lieb internal energy functional. Quasicontinuity and closure properties of the setvalued representing potentials map are also established. The extent of possible nonuniqueness is similar to that found in nonrigorous studies of the conventional theory, namely nonuniqueness can occur for states of collinear magnetization which are eigenstates of S z .

Function space requirements for the singleelectron functions within the multiparticle Schrödinger equation
View Description Hide DescriptionOur previously described method to approximate the manyelectron wavefunction in the multiparticle Schrödinger equation reduces this problem to operations on many singleelectron functions. In this work, we analyze these operations to determine which function spaces are appropriate for various intermediate functions in order to bound the output. This knowledge then allows us to choose the function spaces in which to control the error of a numerical method for singleelectron functions. We find that an efficient choice is to maintain the singleelectron functions in L 2 ∩ L 4, the product of these functions in L 1 ∩ L 2, the Poisson kernel applied to the product in L 4, a function times the Poisson kernel applied to the product in L 2, and the nuclear potential times a function in L 4/3. Due to the integral operator formulation, we do not require differentiability.

Null phase curves and manifolds in geometric phase theory
View Description Hide DescriptionBargmann invariants and null phase curves are known to be important ingredients in understanding the essential nature of the geometric phase in quantum mechanics. Null phase manifolds in quantummechanical ray spaces are submanifolds made up entirely of null phase curves, and so are equally important for geometric phase considerations. It is shown that the complete characterization of null phase manifolds involves both the Riemannian metric structure and the symplectic structure of ray space in equal measure, which thus brings together these two aspects in a natural manner.

Is quantum mechanics exact?
View Description Hide DescriptionWe formulate physically motivated axioms for a physical theory which for systems with a finite number of degrees of freedom uniquely lead to quantum mechanics as the only nontrivial consistent theory. Complex numbers and the existence of the Planck constant common to all systems arise naturally in this approach. The axioms are divided into two groups covering kinematics and basic measurement theory, respectively. We show that even if the second group of axioms is dropped, there are no deformations of quantum mechanics which preserve the kinematic axioms. Thus, any theory going beyond quantum mechanics must represent a radical departure from the usual a priori assumptions about the laws of nature.

Quantum Information and Computation

Convexity of quasientropy type functions: Lieb's and Ando's convexity theorems revisited
View Description Hide DescriptionGiven a positive function f on (0, ∞) and a nonzero real parameter θ, we consider a function in three matrices A, B > 0 and X. This generalizes the notion of monotone metrics on positive definite matrices, and in the literature θ = ±1 has been typical. We investigate how operator monotony of f is sufficient and/or necessary for joint convexity/concavity of . Similar discussions are given for quasientropies and quantum skew informations.

Product formulas for exponentials of commutators
View Description Hide DescriptionWe provide a recursive method for systematically constructing product formula approximations to exponentials of commutators, giving approximations that are accurate to arbitrarily high order. Using these formulas, we show how to approximate unitary exponentials of (possibly nested) commutators using exponentials of the elementary operators, and we upper bound the number of elementary exponentials needed to implement the desired operation within a given error tolerance. By presenting an algorithm for quantum search using evolution according to a commutator, we show that the scaling of the number of exponentials in our product formulas with the evolution time is nearly optimal. Finally, we discuss applications of our product formulas to quantum control and to implementing anticommutators, providing new methods for simulating manybody interaction Hamiltonians.

General Relativity and Gravitation

Metric solutions in torsionless gauge for vacuum conformal gravity
View Description Hide DescriptionIn a recent paper we have established the form of the metrictorsional conformal gravitational field equations, and in the present paper we study their vacuum configurations; we will consider a specific situation that will enable us to look for the torsionless limit: two types of special exact solutions are found eventually. A discussion on general remarks will follow.

Isotropic universe with almost scaleinvariant fourthorder gravity
View Description Hide DescriptionWe study a class of isotropic cosmologies in the fourthorder gravity with Lagrangians of the form L = f(R) + k(G) where R and G are the Ricci and GaussBonnet scalars, respectively. A general discussion is given on the conditions under which this gravitational Lagrangian is scaleinvariant or almost scaleinvariant. We then apply this general background to the specific case L = αR 2 + β Gln G with constants α, β. We find closed form cosmological solutions for this case. One interesting feature of this choice of f(R) and k(G) is that for very small negative value of the parameter β, the Lagrangian L = R 2/3 + βGln G leads to the replacement of the exact de Sitter solution coming from L = R 2 (which is a local attractor) to an exact, powerlaw inflation solution a(t) = t p = t −3/β which is also a local attractor. This shows how one can modify the dynamics from de Sitter to powerlaw inflation by the addition of a Gln Gterm.

Contractions of AdS brane algebra and superGalileon Lagrangians
View Description Hide DescriptionWe examine AdS Galileon Lagrangians using the method of nonlinear realization. By contractions (1) flat curvature limit, (2) nonrelativistic brane algebra limit, and (3) (1) + (2) limits we obtain DBI, NewtonHoock, and Galilean Galileons, respectively. We make clear how these Lagrangians appear as invariant 4forms and/or pseudoinvariant WessZumino (WZ) terms using MaurerCartan (MC) equations on the coset G/SO(3, 1). We show the equations of motion are written in terms of the MC forms only and explain why the inverse Higgs condition is obtained as the equation of motion for all cases. The supersymmetric extension is also examined using a supercoset SU(2, 21)/(SO(3, 1) × U(1)) and five WZ forms are constructed. They are reduced to the corresponding five Galileon WZ forms in the bosonic limit and are candidates for supersymmetric Galileon action.

Covariant differential identities and conservation laws in metrictorsion theories of gravitation. I. General consideration
View Description Hide DescriptionArbitrary diffeomorphically invariant metrictorsion theories of gravity are considered. It is assumed that Lagrangians of such theories contain derivatives of field variables (tensor densities of arbitrary ranks and weights) up to a second order only. The generalized KleinNoether methods for constructing manifestly covariant identities and conserved quantities are developed. Manifestly covariant expressions are constructed without including auxiliary structures like a background metric. In the RiemannCartan space, the following manifestly generally covariant results are presented: (a) The complete generalized system of differential identities (the KleinNoether identities) is obtained. (b) The generalized currents of three types depending on an arbitrary vector field displacements are constructed: they are the canonical Noether current, symmetrized Belinfante current, and identically conserved HilbertBergmann current. In particular, it is stated that the symmetrized Belinfante current does not depend on divergences in the Lagrangian. (c) The generalized boundary Klein theorem (third Noether theorem) is proved. (d) The construction of the generalized superpotential is presented in detail, and questions related to its ambiguities are analyzed.

Dynamical Systems

Qualitative analysis of the anisotropic twobody problem with relativistic potential
View Description Hide DescriptionIn this paper we study the twobody problem that describes the motion of twopoint masses in an anisotropic space under the influence of a Newtonian forcelaw with two relativistic correction terms. We will show that the set of initial conditions leading to collisions and ejections have positive measure and study the capture and escape solutions in the zeroenergy case using the infinity manifold. We will also apply the Melnikov method to show that the flow on the zeroenergy manifold of another potential which is the sum of the classical Keplerian potential and two anisotropic perturbation which also take into account two relativistic correction terms is chaotic.

Classical Mechanics and Classical Fields

Identities from infinitedimensional symmetries of Herglotz variational functional
View Description Hide DescriptionThis paper formulates and proves a theorem which provides the identities corresponding to infinitedimensional symmetry groups of the functional defined by the generalized variational principle of Herglotz in the case of several independent variables. It contains the classical second Noether theorem as a special case. The equations satisfied by the extrema of this functional, when the Lagrangian density depends on first and second order partial derivatives of the argument functions, are found. We apply the theorem to find two new identities satisfied by the fourpotential of the electromagnetic field propagating in a conductive medium. We also obtain an identity corresponding to a gauge symmetry of the KleinGordon equation with dissipation/generation. This identity becomes a continuity law when the wave function is a solution of this equation.
