Complex extension of the representation of the symplectic group associated with the canonical commutation relations

The aim of the investigation is to extend the representation of the real symplectic group associated with the canonical commutation relations into the complex symplectic group. It is shown that an extension exists to a semigroup S such that Sp(2n, R) ? S ? Sp(2n, C). The construction of the extension is achieved by extensive use of Bargmann's reproducing kernel space. We are able to give a simple geometric model for the semigroup S: it is exactly the semigroup of all complex symplectic transformations which increase the U(n, n) “length”.     

Semigroup of positive maps for qudit states and entanglement in tomographic probability representation

Stochastic and bistochastic matrices providing positive maps for spin states (for qudits) are shown to form semigroups with dense intersection with the Lie groups IGL(n,R) and GL(n,R) respectively. The density matrix of a qudit state is shown to be described by a spin tomogram determined by an orbit of the bistochastic semigroup acting on a simplex. A class of positive maps acting transitively on quantum states is introduced by relating stochastic and quantum stochastic maps in the tomographic setting. Finally, the entangled states of two qubits and Bell inequalities are given in the framework of the tomographic probability representation using the stochastic semigroup properties.     

Similarity reductions and new nonlinear exact solutions for the 2D incompressible Euler equations

For the 2D and 3D Euler equations, their existing exact solutions are often in linear form with respect to variables x, y, z. In this paper, the Clarkson–Kruskal reduction method is applied to reduce the 2D incompressible Euler equations to a system of completely solvable ordinary equations, from which several novel nonlinear exact solutions with respect to the variables x and y are found.     

The definition of the product δ(x)χx-α (α>0, real) and its applications

We introduce new space of continuous test functions S? having at x=0 an asymptotic expansion in Dirichlet series. It is shown that on the space S? the product δ(x)χx^-α(α>0, real) is a local linear continuous functional, proportional to the derivative of continuous order δ^(α) of Dirac delta. It is shown that the product δ(x)χx^-α occurs in quantum field theory asymptotically invariant with respect to the dilatation group, if we differentiate causal or retarded propagators.     

Optimized implementations of rational approximations for the Voigt and complex error function

Rational functions are frequently used as efficient yet accurate numerical approximations for real and complex valued functions. For the complex error function w(x+iy), whose real part is the Voigt function K(x,y), code optimizations of rational approximations are investigated. An assessment of requirements for atmospheric radiative transfer modeling indicates a y range over many orders of magnitude and accuracy better than 10⁻⁴. Following a brief survey of complex error function algorithms in general and rational function approximations in particular the problems associated with subdivisions of the x, y plane (i.e., conditional branches in the code) are discussed and practical aspects of Fortran and Python implementations are considered. Benchmark tests of a variety of algorithms demonstrate that programming language, compiler choice, and implementation details influence computational speed and there is no unique ranking of algorithms. A new implementation, based on subdivision of the upper half-plane in only two regions, combining Weideman's rational approximation for small |x|+y<15 and Humlicek's rational approximation otherwise is shown to be efficient and accurate for all x, y.     

Hills or hollows? — An ambiguity in the determination of surface corrugation

The diffraction intensities from a corrugated hard wall are shown to be the same for a given corrugation function ζ(x, y) and ?ζ(?x, ?y), provided that coupling to evanescent waves is negligible. To illustrate the consequences of this ambiguity for both surface structure and surface bonding, the corrugation of the adsorbate system Ni(110) + H(1 × 2) determined from Hediffraction measurements is considered. Possible conditions under which the ambiguity could be resolved are discussed.     

Overdamped motion of interacting particles in general confining potentials: time-dependent and stationary-state analyses

By comparing numerical and analytical results, it is shown that a system of interacting particles under overdamped motion is very well described by a nonlinear Fokker-Planck equation, which can be associated with nonextensive statistical mechanics. The particle-particle interactions considered are repulsive, motivated by three different physical situations: (i) modified Bessel function, commonly used in vortex-vortex interactions, relevant for the flux-front penetration in disordered type-II superconductors; (ii) Yukawa-like forces, useful for charged particles in plasma, or colloidal suspensions; (iii) derived from a Gaussian potential, common in complex fluids, like polymer chains dispersed in a solvent. Moreover, the system is subjected to a general confining potential, ??(x)?=?(??|x| ^z )/z (???>?0, z?>?1), so that a stationary state is reached after a sufficiently long time. Recent numerical and analytical investigations, considering interactions of type (i) and a harmonic confining potential (z?=?2), have shown strong evidence that a q-Gaussian distribution, P(x,t), with q?=?0, describes appropriately the particle positions during their time evolution, as well as in their stationary state. Herein we reinforce further the connection with nonextensive statistical mechanics, by presenting numerical evidence showing that: (a) in the case z?=?2, different particle-particle interactions only modify the diffusion parameter D of the nonlinear Fokker-Planck equation; (b) for z????2, all cases investigated fit well the analytical stationary solution P _st(x), given in terms of a q-exponential (with the same index q?=?0) of the general external potential ??(x). In this later case, we propose an approximate time-dependent P(x,t) (not known analytically for z????2), which is in very good agreement with the simulations for a large range of times, including the approach to the stationary state. The present work suggests that a wide variety of physical phenomena, characterized by repulsive interacting particles under overdamped motion, present a universal behavior, in the sense that all of them are associated with the same entropic form and nonlinear Fokker-Planck equation.     

On the nature of high values of low-temperature heat capacity in the vicinity of critical concentration for ferromagnetism

The results of a study of low-temperature heat capacity of the Mn₂₀Fe_xNi_80?x, Fe₅₀Ni_50?xMn_x, Fe₆₅Ni_35?xCr_x, Cr₁₀Fe_xNi_0?xquasi-binary alloys are compared with the form of relevant magnetic phase diagrams. It is shown that the anomalously high values of a temperature-linear contribution to heat capacity in alloys, lying in the vicinity of a critical concentration of the change of the type of magnetic order, are determined by the presence of “cluster spin glass”.     

Magnetic polaritons in four-sublattice magnetic systems of the GzFx type

Magnetic polaritons in four-sublattice magnetic compounds of the space symmetry D¹⁶_2h with magnetic ordering of the G_zF_x and G_z types, like KMnF₃, RbFeF₃ etc., are considered. Dispersion curves of mixed states of photons and all four magnetic modes are given for some peculiar directions of propagation. The interaction between photons and high-frequency (optic) magnetic modes is shown to be significantly weaker than that between photons and low-frequency spin waves.     

Discrete breathers in deformed graphene

The linear and nonlinear dynamics of elastically deformed graphene have been studied. The region of the stability of a planar graphene sheet has been represented in the space of the two-dimensional strain (? _xx , ? _yy ) with the x and y axes oriented in the zigzag and armchair directions, respectively. It has been shown that the gap in the phonon spectrum appears in graphene under uniaxial deformation in the zigzag or armchair direction, while the gap is not formed under a hydrostatic load. It has been found that graphene deformed uniaxially in the zigzag direction supports the existence of spatially localized nonlinear modes in the form of discrete breathers, the frequency of which decreases with an increase in the amplitude. This indicates soft nonlinearity in the system. It is unusual that discrete breather has frequency within the phonon spectrum of graphene. This is explained by the fact that the oscillation of the discrete breather is polarized in the plane of the graphene sheet, while the phonon spectral band where the discrete breather frequency is located contains phonons oscillating out of plane. The stability of the discrete breather with respect to the small out-of-plane perturbation of the graphene sheet has been demonstrated.     

Boundary Quantum Field Theory on the Interior of the Lorentz Hyperboloid

We construct local, boost covariant boundary QFT nets of von Neumann algebras on the interior of the Lorentz hyperboloid \({\mathfrak{H}_R}\), x ² ? t ² > R ², x > 0, in the two-dimensional Minkowski spacetime. Our first construction is canonical, starting with a local conformal net on \({\mathbb{R}}\), and is analogous to our previous construction of local boundary CFT nets on the Minkowski half-space. This net is in a thermal state at Hawking temperature. Then, inspired by a recent construction by E. Witten and one of us, we consider a unitary semigroup that we use to build up infinitely many nets. Surprisingly, the one-particle semigroup is again isomorphic to the semigroup of symmetric inner functions of the disk. In particular, by considering the U(1)-current net, we can associate with any given symmetric inner function a local, boundary QFT net on \({\mathfrak{H}_R}\). By considering different states, we shall also have nets in a ground state, rather than in a KMS state.     

SUSY-hierarchy of one-dimensional reflectionless potentials

A class of one-dimensional reflectionless potentials is studied. It is found that all possible types of the reflectionless potentials can be combined into one SUSY-hierarchy with a constant potential. An approach for determination of a general form of the reflectionless potential on the basis of construction of such a hierarchy by the recurrent method is proposed. A general integral form of interdependence between superpotentials with neighboring numbers of this hierarchy, opening a possibility to find new reflectionless potentials, is found and has a simple analytical view. It is supposed that any possible type of the reflectionless potential can be expressed through finite number of elementary functions (unlike some presentations of the reflectionless potentials, which are constructed on the basis of soliton solutions or are shape invariant in one or many steps with involving scaling of parameters, and are expressed through series). An analysis of absolute transparency existence for the potential which has the inverse power dependence on space coordinate (and here tunneling is possible), i.e., which has the form V (x) = ± α/|x−x₀|ⁿ (where α and x₀ are constants, n is natural number), is fulfilled. It is shown that such a potential can be reflectionless at n = 2 only. A SUSY-hierarchy of the inverse power reflectionless potentials is constructed. Isospectral expansions of this hierarchy are analyzed.     

Structure of the intermediate high-pressure phases of ternary lead tellurides

In chambers with diamond anvils, the structure of high-pressure phases of ternary lead tellurides Pb_1?x Sn_xTe (x = 0.29) and Pb_1?x Mn_xTe (x = 0.05) and nonstoichiometric crystals Pb_0.55Te_0.45, Pb_0.45Te_0.55 is analyzed by the synchrotron radiation diffraction method at pressures of P up to 14 GPa. The orthorhombic structure of the intermediate high-pressure phase (space group Pnma) is determined for all the samples above 6 GPa. Models of the phase transition in PbTe from the initial rock salt structure to the orthorhombic phase, which constitutes a distorted variant of NaCl, as well as the properties of this phase, are discussed.     

Theory of the period-doubling phenomenon of one-dimensional mappings based on the parameter dependence

A theory is presented of the period-doubling phenomenon of one-dimensional mappings of the form x_n+1 = F(x_n, r), which is different from that of Feigenbaum mainly in that it is based on the r dependence of various quantities rather than on their x dependence. Consequently, it enables us to evaluate, for example, the Lyapunov numbers of periodic orbits as a function of r as well as the Feigenbaum ratio. It is shown that the results of our theory are in good agreement with those of numerical simulations.     

Convergence to SPDEs in Stratonovich Form

We consider the perturbation of parabolic operators of the form ∂ _t  + P(x, D) by large-amplitude highly oscillatory spatially dependent potentials modeled as Gaussian random fields. The amplitude of the potential is chosen so that the solution to the random equation is affected by the randomness at the leading order. We show that, when the dimension is smaller than the order of the elliptic pseudo-differential operator P(x, D), the perturbed parabolic equation admits a solution given by a Duhamel expansion. Moreover, as the correlation length of the potential vanishes, we show that the latter solution converges in distribution to the solution of a stochastic parabolic equation with multiplicative noise that should be interpreted in the Stratonovich sense. The theory of mild solutions for such stochastic partial differential equations is developed. The behavior described above should be contrasted to the case of dimensions larger than or equal to the order of the elliptic pseudo-differential operator P(x, D). In the latter case, the solution to the random equation converges strongly to the solution of a homogenized (deterministic) parabolic equation as is shown in [2]. A stochastic limit is obtained only for sufficiently small space dimensions in this class of parabolic problems.