Volume 57, Issue 9, September 2016

Many physical systems can be described by eigenvalues of nonlinear equations and bifurcation problems with a linear part that is nonselfadjoint, e.g., due to the presence of loss and gain. The balance of these effects is reflected in an antilinear symmetry, e.g., the symmetry. Under the symmetry we show that the nonlinear eigenvalues bifurcating from real linear eigenvalues remain real and the corresponding nonlinear eigenfunctions remain symmetric. The abstract result is applied in a number of physical models of BoseEinstein condensation, nonlinear optics, and superconductivity, and numerical examples are presented.
 ARTICLES

 Partial Differential Equations

The 1D parabolicparabolic PatlakKellerSegel model of chemotaxis: The particular integrable case and soliton solution
View Description Hide DescriptionIn this paper, we investigate the onedimensional parabolicparabolic PatlakKellerSegel model of chemotaxis. For the case when the diffusion coefficient of chemical substance is equal to two, in terms of travelling wave variables the reduced system appears integrable and allows the analytical solution. We obtain the exact soliton solutions, one of which is exactly the onesoliton solution of the Kortewegde Vries equation.

Existence of multibump solutions for the fractional SchrödingerPoisson system
View Description Hide DescriptionThis paper considers the fractional SchrödingerPoisson system in ℝ^{3}. We prove that the problem has mbump solutions under some given conditions which are given in the Introduction. Moreover, the system has more and more multibump solutions as ϵ → 0.
 Representation Theory and Algebraic Methods

Combinatorial bases of basic modules for affine Lie algebras
View Description Hide DescriptionLepowsky and Wilson initiated the approach to combinatorial RogersRamanujan type identities via vertex operator constructions of standard (i.e., integrable highest weight) representations of affine KacMoody Lie algebras. Meurman and Primc developed further this approach for by using vertex operator algebras and Verma modules. In this paper, we use the same method to construct combinatorial bases of basic modules for affine Lie algebras of type and, as a consequence, we obtain a series of RogersRamanujan type identities. A major new insight is a combinatorial parametrization of leading terms of defining relations for level one standard modules for affine Lie algebra of type .

Twoparameter twisted quantum affine algebras
View Description Hide DescriptionWe establish Drinfeld realization for the twoparameter twisted quantum affine algebras using a new method. The Hopf algebra structure for Drinfeld generators is given for both untwisted and twisted twoparameter quantum affine algebras, which include the quantum affine algebras as special cases.

Deformation of noncommutative quantum mechanics
View Description Hide DescriptionIn this paper, the Lie group , of which the kinematical symmetry group G NC of noncommutative quantum mechanics (NCQM) is a special case due to fixed nonzero α, β, and γ, is threeparameter deformation quantized using the method suggested by Ballesteros and Musso [J. Phys. A: Math. Theor. 46, 195203 (2013)]. A certain family of QUE algebras, corresponding to with two of the deformation parameters approaching zero, is found to be in agreement with the existing results of the literature on quantum Heisenberg group. Finally, we dualize the underlying QUE algebra to obtain an expression for the underlying starproduct between smooth functions on .

sl(2)modules by sl(2)coherent states
View Description Hide DescriptionIrreducible sp(4)module with highest weight, labeled by the azimuthal and magnetic quantum numbers l and m, is split into the direct sums of the irreducible su(2) and su(1, 1)submodules in four different ways: finite integer unitary irreducible subspaces corresponding to the orbital angular momentum algebra su(2), infinite positive discrete series of su(1, 1) with an arbitrary halfinteger Bargmann index, and the positive and negative discrete series of su(1, 1) with both the Bargmann indices 1/4 and 3/4. Even and odd coherent states for the positive su(1, 1)submodules with the Bargmann indices 1/4 and 3/4 are constructed and it is shown that they enjoy the property of completeness by two appropriate positive definite measures. We show that the even and odd coherent states themselves form the positive discrete series of su(1, 1) with the Bargmann indices 1/4 and 3/4, respectively. For these even and odd coherent states, we consider the uncertainty relations for the x and ycomponents of the angular momentum as well as the generators of the negative discrete series of su(1, 1) with the Bargmann indices 1/4 and 3/4.
 ManyBody and Condensed Matter Physics

Local gap threshold for frustrationfree spin systems
View Description Hide DescriptionWe improve Knabe’s spectral gap bound for frustrationfree translationinvariant local Hamiltonians in 1D. The bound is based on a relationship between global and local gaps. The global gap is the spectral gap of a sizem chain with periodic boundary conditions, while the local gap is that of a subchain of size n < m with open boundary conditions. Knabe proved that if the local gap is larger than the threshold value 1/(n − 1) for some n > 2, then the global gap is lower bounded by a positive constant in the thermodynamic limit m → ∞. Here we improve the threshold to , which is better (smaller) for all n > 3 and which is asymptotically optimal. As a corollary we establish a surprising fact about 1D translationinvariant frustrationfree systems that are gapless in the thermodynamic limit: for any such system the spectral gap of a sizen chain with open boundary conditions is upper bounded as O(n ^{−2}). This contrasts with gapless frustrated systems where the gap can be Θ(n ^{−1}). It also limits the extent to which the area law is violated in these frustrationfree systems, since it implies that the halfchain entanglement entropy is as a function of spectral gap ϵ. We extend our results to frustrationfree systems on a 2D square lattice.
 Quantum Mechanics

A Fermi golden rule for quantum graphs
View Description Hide DescriptionWe present a Fermi golden rule giving rates of decay of states obtained by perturbing embedded eigenvalues of a quantum graph. To illustrate the procedure in a notationally simpler setting, we first describe a Fermi golden rule for boundary value problems on surfaces with constant curvature cusps. We also provide a resonance existence result which is uniform on compact sets of energies and metric graphs. The results are illustrated by numerical experiments.

Dynamics of uncertainties for bound onedimensional semiclassical wave packets
View Description Hide DescriptionWe study the time evolution of the uncertainties Δx and Δp in position and momentum, respectively, associated with the semiclassical propagation of certain Gaussian initial states. We show that these quantities behave generically as , where P 1 and P 2 are periodic in time with period that of an underlying classical trajectory. We also show that, despite the overall (generically quadratic) growth in time, the uncertainty product ΔxΔp achieves its minimum of ħ/2 at arbitrarily large times.

Entropy and information of a spinless charged particle in timevarying magnetic fields
View Description Hide DescriptionWe calculate the Fisher information (F r and F p) and the Shannon entropies (S r and S p) of a spinless charged particle in three different configurations of timevarying magnetic field, B(t). To do so, we first obtain the exact wave functions for a general timedependent system by using a dynamical invariant method. By considering the solutions for n = m = 0, we were able to obtain the expressions of F r, F p, S r, and S p in terms of a cnumber quantity, ρ, which has to be a real solution of the Milne–Pinney equation. We observe that the inequality F r F p ≤ 16 holds for the systems considered. We also observed squeezing phenomenon in momentum or/and coordinate spaces with increasing time.

Quadratic algebra for superintegrable monopole system in a TaubNUT space
View Description Hide DescriptionWe introduce a Hartmann system in the generalized TaubNUT space with Abelian monopole interaction. This quantum system includes well known KaluzaKlein monopole and MICZwanziger monopole as special cases. It is shown that the corresponding Schrödinger equation of the Hamiltonian is separable in both spherical and parabolic coordinates. We obtain the integrals of motion of this superintegrable model and construct the quadratic algebra and Casimir operator. This algebra can be realized in terms of a deformed oscillator algebra and has finite dimensional unitary representations (unirreps) which provide energy spectra of the system. This result coincides with the physical spectra obtained from the separation of variables.
 Quantum Information and Computation

Bipartite depolarizing maps
View Description Hide DescriptionWe introduce a 3parameter class of maps (1) acting on a bipartite system which are a natural generalisation of the depolarizing channel (and include it as a special case). Then, we find the exact regions of the parameter space that alternatively determine a positive, completely positive, entanglementbreaking, or entanglementannihilating map. This model displays a much richer behaviour than the one shown by a simple depolarizing channel, yet it stays exactly solvable. As an example of this richness, positive partial transposition but not entanglementbreaking maps is found in Theorem 2. A simple example of a positive yet indecomposable map is provided (see the Remark at the end of Section IV). The study of the entanglementannihilating property is fully addressed by Theorem 7. Finally, we apply our results to solve the problem of the entanglement annihilation caused in a bipartite system by a tensor product of local depolarizing channels. In this context, a conjecture posed in the work of Filippov [J. Russ. Laser Res. 35, 484 (2014)] is affirmatively answered, and the gaps that the imperfect bounds of Filippov and Ziman [Phys. Rev. A 88, 032316 (2013)] left open are closed. To arrive at this result, we furthermore show how the Hadamard product between quantum states can be implemented via local operations.

Maximum privacy without coherence, zeroerror
View Description Hide DescriptionWe study the possible difference between the quantum and the private capacities of a quantum channel in the zeroerror setting. For a family of channels introduced by Leung et al. [Phys. Rev. Lett. 113, 030512 (2014)], we demonstrate an extreme difference: the zeroerror quantum capacity is zero, whereas the zeroerror private capacity is maximum given the quantum output dimension.

A universal property for sequential measurement
View Description Hide DescriptionWe study the sequential product the operation on the set of effects, [0, 1]𝒜, of a von Neumann algebra 𝒜 that represents sequential measurement of first p and then q. In their work [J. Math. Phys. 49(5), 052106 (2008)], Gudder and Latrémolière give a list of axioms based on physical grounds that completely determines the sequential product on a von Neumann algebra of type I, that is, a von Neumann algebra ℬ(ℋ) of all bounded operators on some Hilbert space ℋ. In this paper we give a list of axioms that completely determines the sequential product on all von Neumann algebras simultaneously (Theorem 4).
 Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory

The perturbative approach to path integrals: A succinct mathematical treatment
View Description Hide DescriptionWe study finitedimensional integrals in a way that elucidates the mathematical meaning behind the formal manipulations of path integrals occurring in quantum field theory. This involves a proper understanding of how Wick’s theorem allows one to evaluate integrals perturbatively, i.e., as a series expansion in a formal parameter irrespective of convergence properties. We establish invariance properties of such a Wick expansion under coordinate changes and the action of a Lie group of symmetries, and we use this to study essential features of path integral manipulations, including coordinate changes, Ward identities, SchwingerDyson equations, FaddeevPopov gaugefixing, and eliminating fields by their equation of motion. We also discuss the asymptotic nature of the Wick expansion and the implications this has for defining path integrals perturbatively and nonperturbatively.
 General Relativity and Gravitation

Riccati equations for bounded radiating systems
View Description Hide DescriptionWe systematically analyze the nonlinear partial differential equation that determines the behaviour of a bounded radiating spherical mass in general relativity. Four categories of solution are possible. These are identified in terms of restrictions on the gravitational potentials. One category of solution can be related to the horizon function transformation which was recently introduced. A Lie symmetry analysis of the resulting Riccati equation shows that several new classes of exact solutions are possible. The relationship between the horizon function, Euclidean star models, and other earlier investigations is clarified.

Monotoneshort solutions of the TolmanOppenheimerVolkoffde Sitter equation
View Description Hide DescriptionIt is known that spherically symmetric static solutions of the Einstein equations with a positive cosmological constant for the energymomentum tensor of a barotropic perfect fluid are governed by the TolmanOppenheimerVolkoffde Sitter equation. Some sufficient conditions for the existence of monotoneshort solutions (with finite radii) of the equation are given in this article. Then we show that the interior metric can extend to the exterior Schwarzschildde Sitter metric on the exterior vacuum region with twice continuous differentiability. In addition, we investigate the analytic property of the solutions at the vacuum boundary. Our result (Theorem 1) can be considered as the de Sitter version of the result by Rendall and Schmidt [Classical Quantum Gravity 8, 9851000 (1991)]. Furthermore, one can see that there are different properties of the solutions with those of the TolmanOppenheimerVolkoff equation (with zero cosmological constant) in certain situation.

Spaces of paths and the path topology
View Description Hide DescriptionThe natural topology on the space of causal paths of a spacetime depends on the topology chosen on the spacetime itself. Here we consider the effect of using the path topology on spacetime instead of the manifold topology, and its consequences for how properties of spacetime are reflected in the structure of the space of causal paths.

How should spinweighted spherical functions be defined?
View Description Hide DescriptionSpinweighted spherical functions provide a useful tool for analyzing tensorvalued functions on the sphere. A tensor field can be decomposed into complexvalued functions by taking contractions with tangent vectors on the sphere and the normal to the sphere. These component functions are usually presented as functions on the sphere itself, but this requires an implicit choice of distinguished tangent vectors with which to contract. Thus, we may more accurately say that spinweighted spherical functions are functions of both a point on the sphere and a choice of frame in the tangent space at that point. The distinction becomes extremely important when transforming the coordinates in which these functions are expressed, because the implicit choice of frame will also transform. Here, it is proposed that spinweighted spherical functions should be treated as functions on the spin or rotation groups, which simultaneously tracks the point on the sphere and the choice of tangent frame by rotating elements of an orthonormal basis. In practice, the functions simply take a quaternion argument and produce a complex value. This approach more cleanly reflects the geometry involved, and allows for a more elegant description of the behavior of spinweighted functions. In this form, the spinweighted spherical harmonics have simple expressions as elements of the Wigner 𝔇 representations, and transformations under rotation are simple. Two variants of the angularmomentum operator are defined directly in terms of the spin group; one is the standard angularmomentum operator L, while the other is shown to be related to the spinraising operator ð.

Scaleinvariant gauge theories of gravity: Theoretical foundations
View Description Hide DescriptionWe consider the construction of gauge theories of gravity, focussing in particular on the extension of local Poincaré invariance to include invariance under local changes of scale. We work exclusively in terms of finite transformations, which allow for a more transparent interpretation of such theories in terms of gauge fields in Minkowski spacetime. Our approach therefore differs from the usual geometrical description of locally scaleinvariant Poincaré gauge theory (PGT) and Weyl gauge theory (WGT) in terms of Riemann–Cartan and Weyl–Cartan spacetimes, respectively. In particular, we reconsider the interpretation of the Einstein gauge and also the equations of motion of matter fields and test particles in these theories. Inspired by the observation that the PGT and WGT matter actions for the Dirac field and electromagnetic field have more general invariance properties than those imposed by construction, we go on to present a novel alternative to WGT by considering an “extended” form for the transformation law of the rotational gauge field under local dilations, which includes its “normal” transformation law in WGT as a special case. The resulting “extended” Weyl gauge theory (eWGT) has a number of interesting features that we describe in detail. In particular, we present a new scaleinvariant gauge theory of gravity that accommodates ordinary matter and is defined by the most general parityinvariant eWGT Lagrangian that is at most quadratic in the eWGT field strengths, and we derive its field equations. We also consider the construction of PGTs that are invariant under local dilations assuming either the “normal” or “extended” transformation law for the rotational gauge field, but show that they are special cases of WGT and eWGT, respectively.