Volume 49, Issue 9, September 2008
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


The discrete Feynman integral
View Description Hide DescriptionWe construct a genuine Radon measure with values in on the set of paths in representing Feynman’s integral for the discrete Laplacian on , and we prove the Feynman integral formula for the solutions of the Schrödinger equation with Hamiltonian , where is the discrete Laplacian and is an arbitrary bounded potential.

Classification and time evolution of density matrices for a state system
View Description Hide DescriptionA possible method of classifying all density matrices for a state quantum system into different types, using the sameness of the characteristic polynomial of density matrices belonging to a given type, is discussed. The method employs the generator expansion of density matrices. It is demonstrated that density matrices belonging to a given type, although different, share many common properties including the same eigenvalues, the same shape of the region in the parametric space in which lie the allowed parameter values, and the same global averages of the pertinent observables. Furthermore, we discuss the time evolution of a given type of density matrices and show that, in the appropriately defined timedependent comoving matrix basis, a density matrix of a given type belongs, during entire unitary evolution, to the same type. In this way one is able to group and organize the multitude of different density matrices with certain analogous properties, into a smaller number of types, thus systematizing and cataloguing different mixed and pure states of a state system. The general discussion is illustrated throughout with the special case of a state quantum system. Two and threedimensional cross sections of the space of generalized Bloch vectors are determined by the Monte Carlo sampling method for several types of density matrices, providing some insight into the intricate and complex geometric structure of the space of density matrices.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Poisson sigma model with branes and hyperelliptic Riemann surfaces
View Description Hide DescriptionWe derive the explicit form of the superpropagators in the presence of general boundary conditions (coisotropic branes) for the Poisson sigma model. This generalizes the results presented by Cattaneo and Felder [“A path integral approach to the Kontsevich quantization formula,” Commun. Math. Phys.212, 591 (2000)] and Cattaneo and Felder [“Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model,” Lett. Math. Phys.69, 157 (2004)] for Kontsevich’s angle function [Kontsevich, M., “Deformation quantization of Poissonmanifolds I,” eprint arXiv:hep.th/0101170] used in the deformation quantization program of Poissonmanifolds. The relevant superpropagators for branes are defined as gauge fixed homotopy operators of a complex of differential forms on sided polygons with particular “alternating” boundary conditions. In the presence of more than three branes we use first order Riemann theta functions with odd singular characteristics on the Jacobian variety of a hyperelliptic Riemann surface (canonical setting). In genus the superpropagators present zero mode contributions.
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 GENERAL RELATIVITY AND GRAVITATION


Limit curve theorems in Lorentzian geometry
View Description Hide DescriptionThe subject of limit curve theorems in Lorentzian geometry is reviewed. A general limit curve theorem is formulated, which includes the case of converging curves with endpoints and the case in which the limit points assigned since the beginning are one, two, or at most denumerable. Some applications are considered. It is proved that in chronological spacetimes, strong causality is either everywhere verified or everywhere violated on maximizing lightlike segments with open domain. As a consequence, if in a chronological spacetime two distinct lightlike lines intersect each other then strong causality holds at their points. Finally, it is proved that two distinct components of the chronology violating set have disjoint closures or there is a lightlike line passing through each point of the intersection of the corresponding boundaries.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


BiHamiltonian aspects of a matrix Harry Dym hierarchy
View Description Hide DescriptionWe study the Harry Dym hierarchy of nonlinear evolution equations from the biHamiltonian view point. This is done by using the concept of an hierarchy. It allows us to define a matrix Harry Dym hierarchy of commuting Hamiltonian flows in two fields that projects onto the scalar Harry Dym hierarchy. We also show that the conserved densities of the matrix Harry Dym equation can be found by means of a Riccatitype equation.

Superintegrability of the caged anisotropic oscillator
View Description Hide DescriptionWe study the caged anisotropic harmonic oscillator, which is a new example of a superintegrable or accidentally degenerate Hamiltonian. The potential is that of the harmonic oscillator with rational frequency ratio but additionally with barrier terms describing repulsive forces from the principal planes. This confines the classical motion to a sector bounded by the principal planes, or a cage. In three degrees of freedom, there are five isolating integrals of motion, ensuring that all bound trajectories are closed and strictly periodic. Three of the integrals are quadratic in the momenta, the remaining two are polynomials of order and . In the quantum problem, the eigenstates are multiply degenerate, exhibiting copies of the fundamental pattern of the symmetry group .
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 FLUIDS


The flipover effect in selfsimilar laserinduced plasma expansion
View Description Hide DescriptionWe present a rigorous study of a dynamical model for a nonsymmetric expansion of laserinduced plasma plumes into the vacuum. The model is used in the laser film deposition technique and for remote chemical analysis in the socalled laserinduced breakdown spectroscopy. It defines a particular class of solutions of the hydrodynamics equations when the (plasma) mass density, pressure, and temperature as functions of position have level surfaces that are ellipsoids. The time evolution of ellipsoid semiaxes is determined by the dynamical model. In this model we investigate the flipover effect: A pancakelike shape of the plasma plume turns into a cigarlike shape and vice versa in due course of its expansion. The effect has been observed in experiments as well as in numerical simulations. In many practical cases, axially symmetric plasma plumes with the adiabatic constant of (ideal gas) are used. For this case we prove that the flipover effect occurs exactly once in the above dynamical model. This rigorous result agrees with the earlier experimental and numerical evidence and, hence, validates a wide applicability of the model.
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 METHODS OF MATHEMATICAL PHYSICS


The transmission property of the discrete Heisenberg ferromagnetic spin chain
View Description Hide DescriptionWe present a mechanism for displaying the transmission property of the discrete Heisenberg ferromagnetic (DHF) spin chain via a geometric approach. With the aid of a discrete nonlinear Schrödingerlike equation which is the discrete gauge equivalent to the DHF, we show that the determination of transmitting coefficients in the transmission problem is always bistable. Thus, a definite algorithm and general stochastic algorithms are presented. A new invariant periodic phenomenon of the nontransmitting behavior for the DHF, with a large probability, is revealed by an adoption of various stochastic algorithms.

Feynman path integrals for the inverse quartic oscillator
View Description Hide DescriptionThe Feynman path integral representation for the weak solution of the Schrödinger equation with an inverse quartic oscillator potential is given in terms of a well defined infinite dimensional oscillatoryintegral. An analytically continued Wiener integral representation for the solution is provided and an explicit description of the quantum dynamics associated with a nonessentially selfadjoint Hamiltonian is given.

Jump processes on leaves of multibranching trees
View Description Hide DescriptionThe adic numbers have found applications in a wide range of diverse fields of scientific research. In some applications the algebraic properties of adics enter as an indispensable ingredient of the theory. Another class of applications has to do with hierarchical treelike systems. In this context the applications are based on the well known correspondence between adics and the trees with branches emerging from every branching point. Then the algebraic structure does not enter and adics are used merely as a labeling system for the tree branches. We introduce a space of sequences denoted by suitable for labeling the trees with varying number of branches emerging from the branching points. We introduce a nonArchimedean metric in and describe the basic topological properties of . We also demonstrate that the known constructions of the stochastic processes on adics carry over to the stochastic processes on and hence on the corresponding trees.

Lieb–Thirring estimates for nonselfadjoint Schrödinger operators
View Description Hide DescriptionFor general nonsymmetric operators , we prove that the moment of order of negative real parts of its eigenvalues is bounded by the moment of order of negative eigenvalues of its symmetric part . As an application, we obtain Lieb–Thirring estimates for nonselfadjoint Schrödinger operators. In particular, we recover recent results by Frank et al. [Lett. Math. Phys.77, 309 (2006)]. We also discuss moment of resonances of Schrödinger selfadjoint operators.

Approximate similarity reduction for singularly perturbed Boussinesq equation via symmetry perturbation and direct method
View Description Hide DescriptionWe investigate the singularly perturbed Boussinesq equation in terms of the approximate symmetry perturbation method and the approximate direct method. The similarity reduction solutions and similarity reduction equations of different orders display formal coincidence for both methods. Series reduction solutions are consequently derived. For the approximate symmetry perturbation method, similarity reduction equations of the zero order are equivalent to the Painlevé IV, Painlevé I, and Weierstrass elliptic equations. For the approximate direct method, similarity reduction equations of the zero order are equivalent to the Painlevé IV, Painlevé II, Painlevé I, or Weierstrass elliptic equations. The approximate direct method yields more general approximate similarity reductions than the approximate symmetry perturbation method.

An extended twodimensional Toda lattice hierarchy and twodimensional Toda lattice with selfconsistent sources
View Description Hide DescriptionWe extend the twodimensional Toda lattice hierarchy (2DTLH) by its squared eigenfunction symmetries. This extended 2DTLH (ex2DTLH) includes the twodimensional Toda lattice equation with selfconsistent sources (2DTLSCS) as its first nontrivial equation. Lax representation of ex2DTLH is also presented. With the help of the Lax representation, we construct a nonautoBäcklund Darboux transformation (DT) for 2DTLSCS by applying the variation of constants to 2DTLSCS autoBäcklund DT. This DT enables us to find many solutions to 2DTLSCS, including solitons, rational solutions, positons, negatons, and complexitons.

On Painlevé VI transcendents related to the Dirac operator on the hyperbolic disk
View Description Hide DescriptionDirac Hamiltonian on the Poincaré disk in the presence of an Aharonov–Bohm flux and a uniform magnetic field admits a oneparameter family of selfadjoint extensions. We determine the spectrum and calculate the resolvent for each element of this family. Explicit expressions for Green’s functions are then used to find Fredholm determinant representations for the tau function of the Dirac operator with two branch points on the Poincaré disk. Isomonodromic deformation theory for the Dirac equation relates this tau function to a oneparameter class of solutions of the Painlevé VI equation with . We analyze longdistance behavior of the tau function, as well as the asymptotics of the corresponding Painlevé VI transcendents as . Considering the limit of flat space, we also obtain a class of solutions of the Painlevé V equation with .

Evaluation of a integral arising in quantum field theory
View Description Hide DescriptionWe analytically evaluate a dilogarithmic integral that is prototypical of volumes of ideal tetrahedra in hyperbolic geometry. We additionally obtain new representations of the Clausen function and the Catalan constant , as well as new relations between sine and Clausen function values.

Nondistributive algebraic structures derived from nonextensive statistical mechanics
View Description Hide DescriptionWe propose a twoparametric nondistributive algebraic structure that follows from logarithm and exponential functions. Properties of generalized operators are analyzed. We also generalize the proposal into a multiparametric structure (generalization of logarithm and exponential functions and the corresponding algebraic operators). All parameter expressions recover generalization when the corresponding . Nonextensive statistical mechanics has been the source of successive generalizations of entropic forms and mathematical structures, in which this work is a consequence.

A new multicomponent CKP hierarchy and solutions
View Description Hide DescriptionBased on the eigenfunction symmetry constraints, we first propose a new multicomponent Ctype subhierarchy for KadomtsevPetviashvili (mcCKP) hierarchy and its Lax representation, which provides an effective way to find two types of new CKP equation with selfconsistent sources along with their related Lax representations. Also it admits reductions to constrained CKP hierarchy and to a dimensional soliton hierarchy with selfconsistent sources, which gives rise to two types of new Kaup–Kuperschmidt (KK) equation with selfconsistent sources and two types of new bidirectional Kaup–Kuperschmidt equation with selfconsistent sources. By using the solutions of the CKP and KK equations and their corresponding eigenfunctions, soliton solutions for the first type of CKP selfconsistent source (CKPSCS) and of KKSCS are constructed by means of method of variation in constant, respectively.

Leibniz algebra deformations of a Lie algebra
View Description Hide DescriptionIn this note we compute Leibniz algebra deformations of the threedimensional Heisenberg Lie algebra and compare it to its Lie deformations. It turns out that there are three extra Leibniz deformations. We also describe a versal Leibniz deformation of with the versal base.

The Lorentz group and its finite field analogs: Local isomorphism and approximation
View Description Hide DescriptionFinite Lorentz groups acting on fourdimensional vector spaces coordinatized by finite fields with a prime number of elements are represented as homomorphic images of countable, rational subgroups of the Lorentz group acting on real fourdimensional spacetime. Bounded subsets of the real Lorentz group are retractable with arbitrary precision to finite subsets of such rational subgroups. These finite retracts correspond, via local isomorphisms, to wellbehaved subsets of Lorentz groups over finite fields. This establishes a relationship of approximation between the real Lorentz group and Lorentz groups over very large finite fields.
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 ERRATA


Erratum: “A Special Class of Rank 10 and 11 Coxeter Groups” [J. Math. Phys.48, 053512 (2007)]
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