http://arxiv.org/ Mathematical Physics (math-ph) updates on the arXiv.org e-print archive en-us 2010-09-02T20:30:00-05:00 www-admin@arxiv.org Mathematical Physics 1901-01-01T00:00+00:00 1 daily http://arxiv.org/icons/sfx.gif http://arxiv.org/ http://arxiv.org/abs/1009.0289 <p>The direct spreading measures of the Laguerre polynomials, which quantify the distribution of its Rakhmanov probability density along the positive real line in various complementary and qualitatively different ways, are investigated. These measures include the familiar root-mean-square or standard deviation and the information-theoretic lengths of Fisher, Renyi and Shannon types. The Fisher length is explicitly given. The Renyi length of order q (such that 2q is a natural number) is also found in terms of the polynomials parameters by means of two error-free computing approaches; one makes use of the Lauricella functions, which is based on the Srivastava-Niukkanen linearization relation of Laguerre polynomials, and another one which utilizes the multivariate Bell polynomials of Combinatorics. The Shannon length cannot be exactly calculated because of its logarithmic-functional form, but its asymptotics is provided and sharp bounds are obtained by use of an information-theoretic optimization procedure. Finally, all these spreading measures are mutually compared and computationally analyzed; in particular, it is found that the apparent quasi-linear relation between the Shannon length and the standard deviation becomes rigorously linear only asymptotically (i.e. for n&gt;&gt;1). </p> P. S&#xe1;nchez-Moreno, D. Manzano, J.S. Dehesa http://arxiv.org/abs/1009.0293 <p>We give an universal algorithm for testing the local unitary equivalence of states for multipartite system with arbitrary dimensions. </p> Adam Sawicki, Marek Ku&#x15b; http://arxiv.org/abs/1009.0372 <p>We introduce in this paper the contractions $\mathfrak{G}_c$ of $n$-Lie (or Filippov) algebras $\mathfrak{G}$ and show that they have a semidirect structure as their $n=2$ Lie algebra counterparts. As an example, we compute the non-trivial contractions of the simple $A_{n+1}$ Filippov algebras. By using the \.In\"on\"u-Wigner and the generalized Weimar-Woods contractions of ordinary Lie algebras, we compare (in the $\mathfrak{G}=A_{n+1}$ simple case) the Lie algebras Lie$\,\mathfrak{G}_c$ (the Lie algebra of inner endomorphisms of $\mathfrak{G}_c$) with certain contractions $(\mathrm{Lie}\,\mathfrak{G})_{IW}$ and $(\mathrm{Lie}\,\mathfrak{G})_{W-W}$ of the Lie algebra Lie$\,\mathfrak{G}$ associated with $\mathfrak{G}$. </p> J.A. de Azcarraga, J.M. Izquierdo, M. Picon http://arxiv.org/abs/1009.0408 <p>We compute the eigenfunctions and eigenvalues of the periodic integrable spin s XXX model using the coordinate Bethe ansatz. To do so, we compute explicitly the Hamiltonian of the model. These results generalize what has been obtained for spin 1/2 and spin 1 chains. </p> N. Crampe, E. Ragoucy, L. Alonzi http://arxiv.org/abs/1009.0431 <p>For a class of nonnegative, range-1 pair potentials in one dimensional continuous space we prove that any classical ground state of lower density &gt;=1 is a tower-lattice, i.e., a lattice formed by towers of particles the heights of which can differ only by one, and the lattice constant is 1. The potential may be flat or may have a cusp at the origin, it can be continuous, but its derivative has a jump at 1. The result is valid on finite intervals or rings of integer length and on the whole line. </p> Andras Suto http://arxiv.org/abs/1009.0437 <p>We present an algorithm for the explicit numerical calculation of SU(N) and SL(N,C) Clebsch-Gordan coefficients, based on the Gelfand-Tsetlin pattern calculus. Our algorithm is well-suited for numerical implementation; we include a computer code in an appendix. Our exposition presumes only familiarity with the representation theory of SU(2). </p> Arne Alex, Matthias Kalus, Alan Huckleberry, Jan von Delft http://arxiv.org/abs/1009.0444 <p>This course is aimed at graduate students in physics in mathematics and designed to give a comprehensive introduction to Weyl quantization and semiclassics via Egorov's theorem. </p> <p>Chapter 2 gives a quick overview of classical and quantum mechanics on R^d. Some mathematical preliminaries concerning Hilbert space theory, operator theory and tempered distributions are detailed in Chapters 3-5. Weyl quantization and semiclassics are the content of Chapters 6 and 7. Finally, an application of Weyl calculus to Born-Oppenheimer systems is discussed in Chapter 8. </p> Max Lein http://arxiv.org/abs/1009.0464 <p>Conditions are given for the second-order linear differential equation P3 y" + P2 y'- P1 y = 0 to have polynomial solutions, where Pn is a polynomial of degree n. Several application of these results to Schroedinger's equation are discussed. Conditions under which the confluent, biconfluent, and the general Heun equation yield polynomial solutions are explicitly given. Some new classes of exactly solvable differential equation are also discussed. The results of this work are expressed in such way as to allow direct use, without preliminary analysis. </p> Hakan Ciftci, Richard L. Hall, Nasser Saad, Ebubekir Dogu http://arxiv.org/abs/1009.0470 <p>We consider the Wigner equation corresponding to a nonlinear Schroedinger evolution of the Hartree type in the semiclassical limit $\hbar\to 0$. Under appropriate assumptions on the initial data and the interaction potential, we show that the Wigner function is close in $L^2$ to its weak limit, the solution of the corresponding Vlasov equation. The strong approximation allows the construction of semiclassical operator-valued observables, approximating their quantum counterparts in Hilbert-Schmidt topology. The proof makes use of a pointwise-positivity manipulation, which seems necessary in working with the $L^2$ norm and the precise form of the nonlinearity. We employ the Husimi function as a pivot between the classical probability density and the Wigner function, which -- as it is well known -- is not pointwise positive in general. </p> A. Athanassoulis, T. Paul, F. Pezzotti, M. Pulvirenti http://arxiv.org/abs/0810.3313 <p>The class of Koch fractals is one of the most interesting families of fractals, and the study of complex networks is a central issue in the scientific community. In this paper, inspired by the famous Koch fractals, we propose a mapping technique converting Koch fractals into a family of deterministic networks, called Koch networks. This novel class of networks incorporates some key properties characterizing a majority of real-life networked systems---a power-law distribution with exponent in the range between 2 and 3, a high clustering coefficient, small diameter and average path length, and degree correlations. Besides, we enumerate the exact numbers of spanning trees, spanning forests, and connected spanning subgraphs in the networks. All these features are obtained exactly according to the proposed generation algorithm of the networks considered. The network representation approach could be used to investigate the complexity of some real-world systems from the perspective of complex networks. </p> Zhongzhi Zhang, Shuyang Gao, Lichao Chen, Shuigeng Zhou, Hongjuan Zhang, Jihong Guan http://arxiv.org/abs/0811.0509 <p>We consider planar quadrangulations with three marked vertices and discuss the geometry of triangles made of three geodesic paths joining them. We also study the geometry of minimal separating loops, i.e. paths of minimal length among all closed paths passing by one of the three vertices and separating the two others in the quadrangulation. We concentrate on the universal scaling limit of large quadrangulations, also known as the Brownian map, where pairs of geodesic paths or minimal separating loops have common parts of non-zero macroscopic length. This is the phenomenon of confluence, which distinguishes the geometry of random quadrangulations from that of smooth surfaces. We characterize the universal probability distribution for the lengths of these common parts. </p> J. Bouttier, E. Guitter http://arxiv.org/abs/0906.4892 <p>We consider quadrangulations with a boundary and derive explicit expressions for the generating functions of these maps with either a marked vertex at a prescribed distance from the boundary, or two boundary vertices at a prescribed mutual distance in the map. For large maps, this yields explicit formulas for the bulk-boundary and boundary-boundary correlators in the various encountered scaling regimes: a small boundary, a dense boundary and a critical boundary regime. The critical boundary regime is characterized by a one-parameter family of scaling functions interpolating between the Brownian map and the Brownian Continuum Random Tree. We discuss the cases of both generic and self-avoiding boundaries, which are shown to share the same universal scaling limit. We finally address the question of the bulk-loop distance statistics in the context of planar quadrangulations equipped with a self-avoiding loop. Here again, a new family of scaling functions describing critical loops is discovered. </p> J. Bouttier, E. Guitter http://arxiv.org/abs/0909.1242 <p>We show how coupling techniques can be used in some metastable systems to prove that mean metastable exit times are almost constant as functions of the starting microscopic configuration within a "metastable set". In the example of the Random Field Curie Weiss model, we show that these ideas can also be used to prove asymptotic exponentiality of normalized metastable escape times. </p> Alessandra Bianchi, Anton Bovier, Dmitry Ioffe http://arxiv.org/abs/0912.0271 <p>The point of the paper is to show some limitations of geometrical optics in the analysis of subwavelength focusing. We analyze the resolution of the image of a line source radiating in the Maxwell fisheye and the Veselago-Pendry slab lens. The former optical medium is deduced from the stereographic projection of a virtual sphere and displays a heterogeneous refractive index n(r) which is proportional to the inverse of 1+r^2. The latter is described by a homogeneous, but negative, refractive index. It has been suggested that the fisheye makes a perfect lens without negative refraction [Leonhardt, Philbin arxiv:<a href="/abs/0805.4778">0805.4778v2</a>]. However, we point out that the definition of super-resolution in such a heterogeneous medium should be computed with respect to the wavelength in a homogenized medium, and it is perhaps more adequate to talk about a conjugate image rather than a perfect image (the former does not necessarily contains the evanescent components of the source). We numerically find that both the Maxwell fisheye and a thick silver slab lens lead to a resolution close to lambda/3 in transverse magnetic polarization (electric field pointing orthogonal to the plane). We note a shift of the image plane in the latter lens. We also observe that two sources lead to multiple secondary images in the former lens, as confirmed from light rays travelling along geodesics of the virtual sphere. We further observe resolutions ranging from lambda/2 to nearly lambda/4 for magnetic dipoles of varying orientations of dipole moments within the fisheye in transverse electric polarization (magnetic field pointing orthogonal to the plane). Finally, we analyse the Eaton lens for which the source and its image are either located within a unit disc of air, or within a corona 1&lt;r&lt;2 with refractive index $n(r)=\sqrt{2/r-1}$. In both cases, the image resolution is about lambda/2. </p> Sebastien Guenneau, Andre Diatta, Ross McPhedran http://arxiv.org/abs/1005.1298 <p>We present two complementary methods, each applicable in a different range, to evaluate the distribution of the lowest eigenvalue of random matrices in a Jacobi ensemble. The first method solves an associated Painleve VI nonlinear differential equation numerically, with suitable initial conditions that we determine. The second method proceeds via constructing the power-series expansion of the Painleve VI function. Our results are applied in a forthcoming paper in which we model the distribution of the first zero above the central point of elliptic curve L-function families of finite conductor and of conjecturally orthogonal symmetry. </p> Eduardo Due&#xf1;ez, Duc Khiem Huynh, Jon P. Keating, Steven J. Miller, Nina C. Snaith http://arxiv.org/abs/1007.3022 <p>Anomalous transport is usually described either by models of continuous time random walks (CTRW) or, otherwise by fractional Fokker-Planck equations. The asymptotic relation between properly scaled CTRW and fractional diffusion process has been worked out via various approaches widely discussed in literature. Here, we focus on a correspondence between CTRWs and time and space fractional diffusion equation stemming from two different methods aimed to accurately approximate anomalous diffusion processes. One of them is the Monte Carlo simulation of uncoupled CTRW with a L\'evy $\alpha$-stable distribution of jumps in space and a one-parameter Mittag-Leffler distribution of waiting times. The other is based on a discretized form of a subordinated Langevin equation in which the physical time defined via the number of subsequent steps of motion is itself a random variable. Both approaches are tested for their numerical performance and verified with known analytical solutions for the Green function of a space-time fractional diffusion equation. </p> Bartlomiej Dybiec, Ewa Gudowska-Nowak http://arxiv.org/abs/1008.5382 <p>We derive the spin-statistics theorem in both relativistic and non-relativistic first-quantized form for local field theories, extending considerably the earlier proofs. Our derivation is based on the representation theories of groups SU(2) and SL(2,C), latter being the universal covering of the Lorentz group. We include theories that have an internal symmetry group. We discuss relation to the standard representations of the Lorentz group and consistency of the non-relativistic limit. We formulate classical Majorana action in SL(2,C) and demonstrate that the failure to write it using the Dirac representation is simply a result of inexact notation. We discuss relation of the theorem to the canonical quantization. We also decouple the Dirac four-spinor representation to separate particle and anti-particle representations and discuss briefly a geometric proof of the CPT theorem. </p> Lauri J. Suoranta http://arxiv.org/abs/0911.5354 <p>We construct complete sets of (open and closed string) covariant coherent state and mass eigenstate vertex operators in bosonic string theory. By minimally extending the standard definition of coherent states so as to include the string theory requirements, we show that the naive construction of the the closed string coherent states requires the existence of a lightlike compactification of spacetime. When the null winding states in the underlying Hilbert space are projected out the resulting vertex operators satisfy the definition of a coherent state and have a classical interpretation. We present explicitly both the covariant and lightcone gauge realization of the resulting states using the DDF map that relates the two. We also identify the corresponding general lightcone gauge classical solutions around which the quantum states are fluctuating. We go on to show that both the covariant gauge coherent vertex operators, the corresponding lightcone gauge coherent states and the classical solutions all share the same mass and angular momenta and conjecture that the covariant and lightcone gauge states are different manifestations of the same state and share identical interactions. This construction can be used to study the evolution of fundamental cosmic strings as predicted by string theory and may also be useful for other applications where massive string vertex operators are of interest. </p> Dimitri Skliros, Mark Hindmarsh http://arxiv.org/abs/0912.2912 <p>A one dimensional exclusion process is introduced where particles hop to a neighbouring vacant site with a rate that depends on the size of the block they belong to. This model is equivalent to a zero range process (ZRP) and shares the same steady state distribution. However positional ordering is lost in this mapping and spatial correlations which depend on position indices can not be calculated directly from the ZRP equivalence. We devise a method to calculate these correlations and illustrate it by reproducing certain known results. Unexplored correlations in some other model systems are also investigated. This method can be generically used to find exact correlations of one dimensional exclusion processes which has ZRP correspondence. </p> Urna Basu, P. K. Mohanty http://arxiv.org/abs/1009.0128 <p>We consider the non-relativistic Hartree model in the gravitational case, i.e. with attractive Coulomb-Newton interaction. For a given mass, we construct stationary states with non-zero temperature by minimizing the corresponding free energy functional. It is proved that minimizers exist if and only if the temperature of the system is below a certain threshold(possibly infinite), which itself depends on the specific choice of the entropy functional. We also investigate whether the corresponding minimizers are mixed or pure quantum states and characterize a positive critical temperature above which mixed states appear. </p> Gonca L. Aki, Jean Dolbeault (CEREMADE), Christof Sparber http://arxiv.org/abs/1009.0163 <p>The paper deals with the semi-classical behaviour of quantum dynamics for a semi-classical completely integrable system with two degrees of freedom near Liouville regular torus. The phenomomenon of wave packet revivals is demonstrated in this article. The framework of this paper is semi-classical analysis (limit :). For the proofs we use standard tools of real analysis, Fourier analysis and basic analytic number theory. </p> Olivier Labl&#xe9;e (IF)