Volume 56, Issue 5, May 2015
Index of content:
We analyze the resurgence properties of finite-dimensional 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 well-known 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 strong-coupling expansions, obtained using analysis of the holomorphy structure of Picard-Fuchs equations.
- Partial Differential Equations
56(2015); http://dx.doi.org/10.1063/1.4919670View Description Hide Description
In this paper, we establish an existence result for a quasilinear Kirchhoff equation, via a sub- and supersolution approach, by using the Minty-Browder’s Theorem for pseudomonotone operators theory.
56(2015); http://dx.doi.org/10.1063/1.4917284View Description Hide Description
We consider a spatially inhomogeneous sine-Gordon equation with a double-well potential, describing long Josephson junctions with phase-shifts. 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 time-periodic drive is shown to sustain symmetric state, while it damps the antisymmetric one.
56(2015); http://dx.doi.org/10.1063/1.4919542View Description Hide Description
We continue the study of integrability of bi-Hamiltonian systems with a compatible pair of local Poisson structures (H 0, H 1), where H 0 is a strongly skew-adjoint 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.
On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics with horizontal dissipation56(2015); http://dx.doi.org/10.1063/1.4921653View Description Hide Description
In this paper, we investigate the Cauchy problem for the incompressible resistive Hall-magnetohydrodynamic equations with horizontal dissipation. By using very delicate energy estimate, we establish a blow-up criterion for the classical solutions and show the global-in-time existence of the classical solution for small initial data.
56(2015); http://dx.doi.org/10.1063/1.4921637View Description Hide Description
We study a linearly coupled Schrödinger system in ℝ N (N ≤ 3). Assume that the potentials in the system are continuous functions satisfying suitable decay assumptions, but without any symmetry properties, and the parameters in the system satisfy some suitable restrictions. Using the Liapunov-Schmidt reduction methods two times and combining localized energy method, we prove that the problem has infinitely many positive synchronized solutions, which extends result Theorem 1.2 about nonlinearly coupled Schrödinger equations in Ao and Wei [Calculus Var. Partial Differ. Equations 51, 761-798 (2014)] to our linearly coupled problem.
Existence of a least energy nodal solution for a Schrödinger-Kirchhoff equation with potential vanishing at infinity56(2015); http://dx.doi.org/10.1063/1.4921639View Description Hide Description
In this paper, we study a class of nonlocal Schrödinger-Kirchhoff problems involving only continuous functions. Using a minimization argument and a quantitative deformation lemma, we find a least energy nodal (or sign-changing) solution to this problem. Moreover, when the problem presents symmetry, we show that it has infinitely many nontrivial solutions.
- Representation Theory and Algebraic Methods
56(2015); http://dx.doi.org/10.1063/1.4919888View Description Hide Description
We 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.
56(2015); http://dx.doi.org/10.1063/1.4921063View Description Hide Description
In 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 gravito-dyons, we reformulate the generalized linear gravitational field equations of gravito-dyons in terms of split octonion. We express the generalized gravito-Heavisidian (GH) potentials, fields, and various wave equations of gravito-dyons in terms of split octonions variables. Accordingly, we demonstrate the work-energy theorem of classical mechanics reproducing the continuity equation for the case of gravito-dyons in terms of split octonions. Further, we discuss the split octonionic form of linear momentum conservation law for gravito-dyons in the case of linear gravitational theory. We have summarized the various split octonion equations for the case of the generalized GH-field of gravito-dyons and the generalized electromagnetic field of dyons. The unified fields of dyons and gravito-dyons have been demonstrated and corresponding field equations are discussed in unique and consistent manner in terms of split octonions.
56(2015); http://dx.doi.org/10.1063/1.4921382View Description Hide Description
The 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.
- Many-Body and Condensed Matter Physics
56(2015); http://dx.doi.org/10.1063/1.4919967View Description Hide Description
We consider free lattice fermions subjected to a static bounded potential and a time- and space-dependent 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, operator-valued) 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 time-correlation 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 so-called 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 Schrödinger-Einstein-Podolsky-Rosen steering in a class of probabilistic theories56(2015); http://dx.doi.org/10.1063/1.4919546View Description Hide Description
Steering is one of the most counter intuitive non-classical features of bipartite quantum system, first noticed by Schrödinger at the early days of quantum theory. On the other hand, measurement incompatibility is another non-classical 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 non-classical 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 Schrödinger-Einstein-Podolsky-Rosen 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.
56(2015); http://dx.doi.org/10.1063/1.4919674View Description Hide Description
Due to the long-range character of the Coulomb interaction theoretical description of low-energy nuclear reactions with charged particles still remains a formidable task. One way of dealing with the problem in an integral-equation 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 partial-wave 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.
56(2015); http://dx.doi.org/10.1063/1.4921171View Description Hide Description
The nine-dimensional MICZ-Kepler problem has been established recently as a system describing the motion of a nine-dimensional 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 MICZ-Kepler problem is equivalent to a harmonic oscillator via generalized Hurwitz transformation. In this work, exact analytical solutions of the Schrödinger equation for the nine-dimensional MICZ-Kepler problem are successfully built for the first time and the degeneration degree of the energy is also calculated.
56(2015); http://dx.doi.org/10.1063/1.4920924View Description Hide Description
We consider the Pauli operator in ℝ3 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.
56(2015); http://dx.doi.org/10.1063/1.4921227View Description Hide Description
Atomic 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 Hartee-Fock 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.
56(2015); http://dx.doi.org/10.1063/1.4921344View Description Hide Description
We study special classes of potentials for which the one-dimensional (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
56(2015); http://dx.doi.org/10.1063/1.4920923View Description Hide Description
We propose a non-commutative extension of the Pauli stabilizer formalism. The aim is to describe a class of many-body quantum states which is richer than the standard Pauli stabilizer states. In our framework, stabilizer operators are tensor products of single-qubit operators drawn from the group 〈αI, X, S〉, where α = eiπ/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 non-Abelian anyons.
56(2015); http://dx.doi.org/10.1063/1.4921265View Description Hide Description
In 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 non-positive 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 Gaussian-to-Gaussian (not necessarily positive) superoperators in terms of their action on the characteristic function of the inputs. For the special case of one-mode mappings, we also show that any Gaussian-to-Gaussian superoperator can be expressed as a concatenation of a phase-space 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 multi-mode scenario, we prove that it still holds at least under the further hypothesis of homogeneous action on the covariance matrix.
56(2015); http://dx.doi.org/10.1063/1.4921341View Description Hide Description
When 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.