Volume 45, Issue 2, February 2004
Index of content:
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Eigenvalues of Ruijsenaars–Schneider models associated with root system in Bethe ansatz formalism
View Description Hide DescriptionRuijsenaars–Schneider models associated with root system with a discrete coupling constant are studied. The eigenvalues of the Hamiltonian are given in terms of the Bethe ansatz formulas. Taking the “nonrelativistic” limit, we obtain the spectrum of the corresponding Calogero–Moser systems in the third formulas of Felder et al.

Eigenvalue problems of a twodimensional Schrödinger operator with nonparabolic effective mass
View Description Hide DescriptionIn this paper, we study the eigenvalue problem for the Schrödinger operator on a two dimensional disk with nonparabolic effective mass approximation. Here the effective mass depends on the energy states. Our results mainly concern with the number of energy states lying in a wire and the monotonicity of energy states with respect to the depth of the wire.

Transition elements for a nonHermitian quadratic Hamiltonian
View Description Hide DescriptionThe nonHermitian quadratic Hamiltonian is analyzed, where and are harmonic oscillator creation and annihilation operators and ω, α, and β are real constants. For the case that it is shown using operator techniques that the Hamiltonian possesses real and positive eigenvalues. A generalized Bogoliubov transformation allows the energy eigenstates to be constructed from the algebra and states of the harmonic oscillator. The eigenstates are shown to possess an imaginary norm for a large range of the parameter space. Finding the orthonormal dual space allows the inner product to be redefined using the complexification procedure of Bender et al. for nonHermitian Hamiltonians. Transition probabilities governed by are shown to be manifestly unitary when the complexification procedure is followed. A specific transition element between harmonic oscillator states is evaluated for both the Hermitian and nonHermitian cases to identify the differences in time evolution.

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


The Euler characteristic and the first Chern number in the covariant phase space formulation of string theory
View Description Hide DescriptionUsing a covariant description of the geometry of deformations for extendons, it is shown that the topological corrections for the string action associated with the Euler characteristic and the first Chern number of the normal bundle of the world sheet, although do not give dynamics to the string, modify the symplectic properties of the covariant phase space of the theory. Future extensions of the present results are outlined.

Construction of a family of quantum Ornstein–Uhlenbeck semigroups
View Description Hide DescriptionFor a given quasifree state on the CCR algebra over one dimensional Hilbert space, a family of Markovian semigroups which leave the quasifree state invariant is constructed by means of noncommutative elliptic operators and Dirichlet forms on von Neumann algebras. The generators (Dirichlet operators) of the semigroups are analyzed and the spectrums together with eigenspaces are found. When restricted to a maximal Abelian subalgebra, the semigroups are reduced to a unique Markovian semigroup of classical Ornstein–Uhlenbeck process.

Existence of mesons and mass splitting in strong coupling lattice quantum chromodynamics
View Description Hide DescriptionWe consider one flavor lattice quantum chromodynamics in the imaginary time functional integral formulation for space dimensions with Dirac spin matrices, small hopping parameter κ, and zero plaquette coupling. We determine the energymomentum spectrum associated with fourcomponent gauge invariant local meson fields which are composites of a quark and an antiquark field. For the associated correlation functions, we establish a Feynman–Kac formula and a spectral representation. Using this representation, we show that the mass spectrum consists of two distinct masses and given by where is real analytic. For and have multiplicity two and the mass splitting is for one mass has multiplicity one and the other three, with mass splitting In the subspace of the Hilbert space generated by an even number of fermion fields the dispersion curves are isolated (upper gap property) up to near the twomeson threshold of asymptotic mass

 GENERAL RELATIVITY AND GRAVITATION


Warped product approach to universe with nonsmooth scale factor
View Description Hide DescriptionIn the framework of Lorentzian warped products, we study the Friedmann–Robertson–Walker cosmological model to investigate nonsmooth curvatures associated with multiple discontinuities involved in the evolution of the universe. In particular we analyze nonsmooth features of the spatially flat Friedmann–Robertson–Walker universe by introducing double discontinuities occurred at the radiationmatter and matterlambda phase transitions in astrophysical phenomenology.

On the classification of type D space–times
View Description Hide DescriptionWe give a classification of the type D space–times based on the invariant differential properties of the Weyl principal structure. Our classification is established using tensorial invariants of the Weyl tensor and, consequently, besides its intrinsic nature, it is valid for the whole set of the type D metrics and it applies on both, vacuum and nonvacuum solutions. We consider the Cottonzero type D metrics and we study the classes that are compatible with this condition. The subfamily of space–times with constant argument of the Weyl eigenvalue is analyzed in more detail by offering a canonical expression for the metric tensor and by giving a generalization of some results about the nonexistence of purely magnetic solutions. The usefulness of these results is illustrated in characterizing and classifying a family of Einstein–Maxwell solutions. Our approach permits us to give intrinsic and explicit conditions that label every metric, obtaining in this way an operational algorithm to detect them. In particular a characterization of the Reissner–Nordström metric is accomplished.

Existence of initial data satisfying the constraints for the spherically symmetric Einstein–Vlasov–Maxwell system
View Description Hide DescriptionUsing ODE techniques we prove the existence of large classes of initial data satisfying the constraints for the spherically symmetric Einstein–Vlasov–Maxwell system. These include data for which the ratio of total charge to total mass is arbitrarily large.

Relativistic stars in differential rotation: bounds on the dragging rate and on the rotational energy
View Description Hide DescriptionFor general relativistic equilibrium stellar models (stationary axisymmetric asymptotically flat and convectionfree) with differential rotation, it is shown that for a wide class of rotation laws the distribution of angular velocity of the fluid has a sign, say “positive,” and then both the dragging rate and the angular momentum density are positive. In addition, the “mean value” (with respect to an intrinsic density) of the dragging rate is shown to be less than the mean value of the fluid angular velocity (in full general, without having to restrict the rotation law, nor the uniformity in sign of the fluid angular velocity); this inequality yields the positivity and an upper bound of the total rotational energy.

 DYNAMICAL SYSTEMS


Elliptic solitons and Gröbner bases
View Description Hide DescriptionWe consider the solution of spectral problems with elliptic coefficients in the framework of the Hermite Ansatz. We show that the search for exactly solvable potentials and their spectral characteristics is reduced to a system of polynomial equations solvable by the Gröbner bases method and others. New integrable potentials and corresponding solutions of the SawadaKotera, KaupKupershmidt, Boussinesq equations and others are found.

A class of nonautonomous coupled KdV systems
View Description Hide DescriptionA class of nonautonomous coupled Korteweg–de Vries (KdV) systems in dimensions are considered for integrability classification. Integrability of the systems is associated with the existence of a certain recursion operator. Some new integrable nonautonomous twocomponent KdV systems are found.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


2geometries and the Hamilton–Jacobi equation
View Description Hide DescriptionBy using two different procedures we show that on the space of solutions of a certain class of secondorder ordinary differential equations, a twodimensional definite or indefinite metric, can be constructed such that the twodimensional Hamilton–Jacobi equation, holds. Furthermore, we show that this structure is invariant under a certain subset of contact transformations (canonical transformations). Two examples are given.

 STATISTICAL PHYSICS


Analogies between colored Lévy noise and random channel approach to disordered kinetics
View Description Hide DescriptionWe point out some interesting analogies between colored Lévy noise and the random channel approach to disordered kinetics. These analogies are due to the fact that the probability density of the Lévy noise source plays a similar role as the probability density of rate coefficients in disordered kinetics. Although the equations for the two approaches are not identical, the analogies can be used for deriving new, useful results for both problems. The random channel approach makes it possible to generalize the fractional Uhlenbeck–Ornstein processes (FUO) for space and timedependent colored noise. We describe the properties of colored noise in terms of characteristic functionals, which are evaluated by using a generalization of Huber’s approach to complex relaxation [Phys. Rev. B 31, 6070 (1985)]. We start out by investigating the properties of symmetrical white noise and then define the Lévy colored noise in terms of a Langevin equation with a Lévy white noise source. We derive exact analytical expressions for the various characteristic functionals, which characterize the noise, and a functional fractional Fokker–Planck equation for the probability density functional of the noise at a given moment in time. Second, by making an analogy between the theory of colored noise and the random channel approach to disordered kinetics, we derive fractional equations for the evolution of the probability densities of the random rate coefficients in disordered kinetics. These equations serve as a basis for developing methods for the evaluation of the statistical properties of the random rate coefficients from experimental data. Special attention is paid to the analysis of systems for which the observed kinetic curves can be described by linear or nonlinear stretched exponential kinetics.

 METHODS OF MATHEMATICAL PHYSICS


On the integration of products of Whittaker functions with respect to the second index
View Description Hide DescriptionSeveral new formulas are developed that enable the evaluation of a family of definite integrals containing the product of two Whittaker functions. The integration is performed with respect to the second index μ, and the first index κ is permitted to have any complex value, within certain restrictions required for convergence. The method utilizes complex contour integration along with various symmetry relations satisfied by the Whittaker functions. The new results derived in this article are complementary to the previously known integrals of products of Whittaker functions, which generally treat integration with respect to either the first index κ or the primary argument x. A physical application involving radiative transport is discussed.

Topologically nontrivial quantum layers
View Description Hide DescriptionGiven a complete noncompact surface Σ embedded in we consider the Dirichlet Laplacian in the layer Ω that is defined as a tubular neighborhood of constant width about Σ. Using an intrinsic approach to the geometry of Ω, we generalize the spectral results of the original paper by Duclos et al. [Commun. Math. Phys. 223, 13 (2001)] to the situation when Σ does not possess poles. This enables us to consider topologically more complicated layers and state new spectral results. In particular, we are interested in layers built over surfaces with handles or several cylindrically symmetric ends. We also discuss more general regions obtained by compact deformations of certain Ω.

Star exponentials for any ordering of the elements of the inhomogeneous symplectic Lie algebra
View Description Hide DescriptionWe compute for any ordering the star exponentials of all polynomials of degree not greater than two on the 2ldimensional phase space of a quantum system with l degrees of freedom, and we show in the particular case of the Moyal star product that the Weyl transform of the Moyal star exponential of the onedimensional harmonic oscillator Hamiltonian is the evolution operator of this quantum system.

A system of nonlinear algebraic equations connected with the multisoliton solution of the Benjamin–Ono equation
View Description Hide DescriptionThe multisoliton solution of the Benjamin–Ono equation is derived from the system of nonlinear algebraic equations. This finding is unexpected from the scheme of the inverse scattering transform method, which constructs the multisoliton solution through the system of linear algebraic equations. The anlaysis developed here is also applied to the rational multisoliton solution of the Kadomtsev–Petviashvili equation.

On the reduction and the existence of approximate analytic solutions of some basic nonlinear ODEs in mathematical physics and nonlinear mechanics
View Description Hide DescriptionUsing series of admissible functional transformations we reduce the onedimensional axisymmetric nonlinear Schrödinger (NLS) equation, as well as the forced damped nonlinear Duffing (NLD) equation to equivalent nonlinear firstorder integrodifferential equations. The forced undamped (NLD) equation results as a special case. The reduced integrodifferential equations are exact. In the limits of small or large values of the parameters characterizing these nonlinear problems, we prove that further reductions lead to firstorder nonlinear ordinary differential equations which, except in case of the (NLS) equation, are of the Abel classes. The approximate reduced (NLS) equation admits exact analytic solutions. On the other hand, taking into account the known exact analytic solutions of the equivalent Abel classes of equations we show that there do not exist analytic solutions of the above two nonlinear Duffing oscillators. However, if further asymptotic approximations take place, new approximate analytic solutions concerning the (NLD) equations are constructed.

 COMMENTS AND ERRATA


Erratum: “Hall effect in noncommutative coordinates” [J. Math. Phys. 43, 4592 (2002)]
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