Volume 56, Issue 5, May 2015
Index of content:

We analyze the resurgence properties of finitedimensional exponential integrals which are prototypes for partition functions in quantum field theories. In these simple examples, we demonstrate that perturbation theory, even at arbitrarily weak coupling, fails as the argument of the coupling constant is varied. It is wellknown that perturbation theory also fails at stronger coupling. We show that these two failures are actually intimately related. The formalism of resurgent transseries, which takes into account global analytic continuation properties, fixes both problems and provides an arbitrarily accurate description of exact result for any value of coupling. This means that strong coupling results can be deduced by using merely weak coupling data. Finally, we give another perspective on resurgence theory by showing that the monodromy properties of the weak coupling results are in precise agreement with the monodromy properties of the strongcoupling expansions, obtained using analysis of the holomorphy structure of PicardFuchs equations.
 ARTICLES

 Partial Differential Equations

A subsupersolution approach for a quasilinear Kirchhoff equation
View Description Hide DescriptionIn this paper, we establish an existence result for a quasilinear Kirchhoff equation, via a sub and supersolution approach, by using the MintyBrowder’s Theorem for pseudomonotone operators theory.

Decay of bound states in a sineGordon equation with doublewell potentials
View Description Hide DescriptionWe consider a spatially inhomogeneous sineGordon equation with a doublewell potential, describing long Josephson junctions with phaseshifts. We discuss the interactions of symmetric and antisymmetric bound states in the system. Using a multiple scale expansion, we show that the modes decay algebraically in time due to the energy transfer from the discrete to the continuous spectrum. In particular, exciting the two modes at the same time yields an increased decay rate. An external timeperiodic drive is shown to sustain symmetric state, while it damps the antisymmetric one.

On integrability of some biHamiltonian two field systems of partial differential equations
View Description Hide DescriptionWe continue the study of integrability of biHamiltonian systems with a compatible pair of local Poisson structures (H 0, H 1), where H 0 is a strongly skewadjoint operator. This is applied to the construction of some new two field integrable systems of PDE by taking the pair (H 0, H 1) in the family of compatible Poisson structures that arose in the study of cohomology of moduli spaces of curves.
 Representation Theory and Algebraic Methods

A proof of the conjecture by CarpentierDe SoleKac
View Description Hide DescriptionWe prove the following conjecture by Carpentier, De Sole, and Kac: let K be a differential field and R be a differential subring of K. Let M be a matrix whose elements are differential operators with coefficients in R. Then, if M has degeneracy degree 1, the Dieudonné determinant of M lies in R.

Split octonion reformulation of generalized linear gravitational field equations
View Description Hide DescriptionIn this paper, we describe the properties of split octonions and their connection with the 2 × 2 Zorn vector matrix containing both scalar and vector components. Starting with a brief description of gravitodyons, we reformulate the generalized linear gravitational field equations of gravitodyons in terms of split octonion. We express the generalized gravitoHeavisidian (GH) potentials, fields, and various wave equations of gravitodyons in terms of split octonions variables. Accordingly, we demonstrate the workenergy theorem of classical mechanics reproducing the continuity equation for the case of gravitodyons in terms of split octonions. Further, we discuss the split octonionic form of linear momentum conservation law for gravitodyons in the case of linear gravitational theory. We have summarized the various split octonion equations for the case of the generalized GHfield of gravitodyons and the generalized electromagnetic field of dyons. The unified fields of dyons and gravitodyons have been demonstrated and corresponding field equations are discussed in unique and consistent manner in terms of split octonions.

Levi decomposable subalgebras of the symplectic algebra C 2
View Description Hide DescriptionThe semisimple subalgebras of the symplectic algebra C 2 are well known. In this article, we classify the Levi decomposable subalgebras of the symplectic algebra C 2, up to inner automorphism. By Levi’s theorem, a full classification of the subalgebras of C 2 would be complete with a classification of its solvable subalgebras.
 ManyBody and Condensed Matter Physics

Microscopic conductivity of lattice fermions at equilibrium. I. Noninteracting particles
View Description Hide DescriptionWe consider free lattice fermions subjected to a static bounded potential and a time and spacedependent electric field. For any bounded convex region ℛ ⊂ ℝ^{ d } (d ≥ 1) of space, electric fields within drive currents. At leading order, uniformly with respect to the volume of and the particular choice of the static potential, the dependency on of the current is linear and described by a conductivity (tempered, operatorvalued) distribution. Because of the positivity of the heat production, the real part of its Fourier transform is a positive measure, named here (microscopic) conductivity measure of , in accordance with Ohm’s law in Fourier space. This finite measure is the Fourier transform of a timecorrelation function of current fluctuations, i.e., the conductivity distribution satisfies Green–Kubo relations. We additionally show that this measure can also be seen as the boundary value of the Laplace–Fourier transform of a socalled quantum current viscosity. The real and imaginary parts of conductivity distributions are related to each other via the Hilbert transform, i.e., they satisfy Kramers–Kronig relations. At leading order, uniformly with respect to parameters, the heat production is the classical work performed by electric fields on the system in presence of currents. The conductivity measure is uniformly bounded with respect to parameters of the system and it is never the trivial measure 0 dν. Therefore, electric fields generally produce heat in such systems. In fact, the conductivity measure defines a quadratic form in the space of Schwartz functions, the Legendre–Fenchel transform of which describes the resistivity of the system. This leads to Joule’s law, i.e., the heat produced by currents is proportional to the resistivity and the square of currents.
 Quantum Mechanics

Measurement incompatibility and SchrödingerEinsteinPodolskyRosen steering in a class of probabilistic theories
View Description Hide DescriptionSteering is one of the most counter intuitive nonclassical features of bipartite quantum system, first noticed by Schrödinger at the early days of quantum theory. On the other hand, measurement incompatibility is another nonclassical feature of quantum theory, initially pointed out by Bohr. Recently, Quintino et al. [Phys. Rev. Lett. 113, 160402 (2014)] and Uola et al. [Phys. Rev. Lett. 113, 160403 (2014)] have investigated the relation between these two distinct nonclassical features. They have shown that a set of measurements is not jointly measurable (i.e., incompatible) if and only if they can be used for demonstrating SchrödingerEinsteinPodolskyRosen steering. The concept of steering has been generalized for more general abstract tensor product theories rather than just Hilbert space quantum mechanics. In this article, we discuss that the notion of measurement incompatibility can be extended for general probability theories. Further, we show that the connection between steering and measurement incompatibility holds in a border class of tensor product theories rather than just quantum theory.

New approach to folding with the Coulomb wave function
View Description Hide DescriptionDue to the longrange character of the Coulomb interaction theoretical description of lowenergy nuclear reactions with charged particles still remains a formidable task. One way of dealing with the problem in an integralequation approach is to employ a screened Coulomb potential. A general approach without screening requires folding of kernels of the integral equations with the Coulomb wave. A new method of folding a function with the Coulomb partial waves is presented. The partialwave Coulomb function both in the configuration and momentum representations is written in the form of separable series. Each term of the series is represented as a product of a factor depending only on the Coulomb parameter and a function depending on the spatial variable in the configuration space and the momentum variable if the momentum representation is used. Using a trial function, the method is demonstrated to be efficient and reliable.

Exact analytical solutions of the Schrödinger equation for the ninedimensional MICZKepler problem
View Description Hide DescriptionThe ninedimensional MICZKepler problem has been established recently as a system describing the motion of a ninedimensional charged particle in the Coulomb potential with the presence of the SO(8) monopole. Interestingly, this is the last case of dimension in which the MICZKepler problem is equivalent to a harmonic oscillator via generalized Hurwitz transformation. In this work, exact analytical solutions of the Schrödinger equation for the ninedimensional MICZKepler problem are successfully built for the first time and the degeneration degree of the energy is also calculated.

A criterion for the existence of zero modes for the Pauli operator with fastly decaying fields
View Description Hide DescriptionWe consider the Pauli operator in ℝ^{3} for magnetic fields in L ^{3/2} that decay at infinity as with β > 0. In this case, we are able to prove that the existence of a zero mode for this operator is equivalent to a quantity δ(B), defined below, being equal to zero. Complementing a result from Balinsky et al. [J. Phys. A: Math. Gen. 34, L19–L23 (2001)], this implies that for the class of magnetic fields considered, Sobolev, Hardy, and Cwikel, Lieb, Rosenblum (CLR) inequalities hold whenever the magnetic field has no zero mode.

Analytical evaluation of atomic form factors: Application to Rayleigh scattering
View Description Hide DescriptionAtomic form factors are widely used for the characterization of targets and specimens, from crystallography to biology. By using recent mathematical results, here we derive an analytical expression for the atomic form factor within the independent particle model constructed from nonrelativistic screened hydrogenic wave functions. The range of validity of this analytical expression is checked by comparing the analytically obtained form factors with the ones obtained within the HarteeFock method. As an example, we apply our analytical expression for the atomic form factor to evaluate the differential cross section for Rayleigh scattering off neutral atoms.

Potentials of the Heun class: The triconfluent case
View Description Hide DescriptionWe study special classes of potentials for which the onedimensional (or radial) Schrödinger equation can be reduced to a triconfluent Heun equation by a suitable coordinate transformation together with an additional transformation of the wave function. In particular, we analyze the behaviour of those subclasses of the potential arising when the ordinary differential equation governing the coordinate transformation admits explicit analytic solutions in terms of the radial variable. Furthermore, we obtain formulae for solutions of the eigenvalue problem of the associated radial Schrödinger operator. Last but not least, using methods of supersymmetric quantum mechanics we relate the considered potentials to a new class of exactly solvable ones.
 Quantum Information and Computation

A noncommuting stabilizer formalism
View Description Hide DescriptionWe propose a noncommutative extension of the Pauli stabilizer formalism. The aim is to describe a class of manybody quantum states which is richer than the standard Pauli stabilizer states. In our framework, stabilizer operators are tensor products of singlequbit operators drawn from the group 〈αI, X, S〉, where α = e^{iπ/4} and S = diag(1, i). We provide techniques to efficiently compute various properties related to bipartite entanglement, expectation values of local observables, preparation by means of quantum circuits, parent Hamiltonians, etc. We also highlight significant differences compared to the Pauli stabilizer formalism. In particular, we give examples of states in our formalism which cannot arise in the Pauli stabilizer formalism, such as topological models that support nonAbelian anyons.

Normal form decomposition for GaussiantoGaussian superoperators
View Description Hide DescriptionIn this paper, we explore the set of linear maps sending the set of quantum Gaussian states into itself. These maps are in general not positive, a feature which can be exploited as a test to check whether a given quantum state belongs to the convex hull of Gaussian states (if one of the considered maps sends it into a nonpositive operator, the above state is certified not to belong to the set). Generalizing a result known to be valid under the assumption of complete positivity, we provide a characterization of these GaussiantoGaussian (not necessarily positive) superoperators in terms of their action on the characteristic function of the inputs. For the special case of onemode mappings, we also show that any GaussiantoGaussian superoperator can be expressed as a concatenation of a phasespace dilatation, followed by the action of a completely positive Gaussian channel, possibly composed with a transposition. While a similar decomposition is shown to fail in the multimode scenario, we prove that it still holds at least under the further hypothesis of homogeneous action on the covariance matrix.

de Finetti reductions for correlations
View Description Hide DescriptionWhen analysing quantum information processing protocols, one has to deal with large entangled systems, each consisting of many subsystems. To make this analysis feasible, it is often necessary to identify some additional structures. de Finetti theorems provide such a structure for the case where certain symmetries hold. More precisely, they relate states that are invariant under permutations of subsystems to states in which the subsystems are independent of each other. This relation plays an important role in various areas, e.g., in quantum cryptography or state tomography, where permutation invariant systems are ubiquitous. The known de Finetti theorems usually refer to the internal quantum state of a system and depend on its dimension. Here, we prove a different de Finetti theorem where systems are modelled in terms of their statistics under measurements. This is necessary for a large class of applications widely considered today, such as device independent protocols, where the underlying systems and the dimensions are unknown and the entire analysis is based on the observed correlations.
 Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory

MickelssonRajeev cocycle corresponding to dimension five
View Description Hide DescriptionWe construct a MickelssonRajeevtype cocycle corresponding to five space dimensions. In the three dimensional case, this cocycle is shown to be equivalent to the original cocycle proposed by Mickelsson and Rajeev. Furthermore, we construct a local representative for this cocycle and evaluate it explicitly on a five dimensional torus.
 General Relativity and Gravitation

The KastlerKalauWalze type theorem for sixdimensional manifolds with boundary
View Description Hide DescriptionIn this paper, we define lower dimensional volumes of spin manifolds with boundary. We compute the lower dimensional volume for 6dimensional spin manifolds with boundary and derive the gravity on boundary from the noncommutative residue associated with Dirac operators. For 6dimensional manifolds with boundary, we also get a KastlerKalauWalze type theorem for a general fourth order operator.

Generating static perfectfluid solutions of Einstein’s equations
View Description Hide DescriptionWe present a method for generating exact interior solutions of Einstein’s equations in the case of static and axially symmetric perfectfluid spacetimes. The method is based upon a transformation that involves the metric functions as well as the density and pressure of the seed solution. In the limiting vacuum case, it reduces to the ZipoyVoorhees transformation that can be used to generate metrics with multipole moments. All the metric functions of the new solution can be calculated explicitly from the seed solution in a simple manner. The physical properties of the resulting new solutions are shown to be completely different from those of the seed solution.
 Dynamical Systems

Quasiperiodic solutions of nonlinear beam equation with prescribed frequencies
View Description Hide DescriptionConsider the one dimensional nonlinear beam equation utt + uxxxx + mu + u ^{3} = 0 under Dirichlet boundary conditions. We show that for any m > 0 but a set of small Lebesgue measure, the above equation admits a family of smallamplitude quasiperiodic solutions with ndimensional Diophantine frequencies. These Diophantine frequencies are the small dilation of a prescribed Diophantine vector. The proofs are based on an infinite dimensional KolmogorovArnoldMoser iteration procedure and a partial Birkhoff normal form.