Volume 46, Issue 3, March 2005
Index of content:
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Recursive calculation of effective potential and variational resummation
View Description Hide DescriptionWe set up a method for a recursive calculation of the effective potential which is applied to a cubic potential with imaginary coupling. The result is resummed using variational perturbation theory, yielding an exponentially fast convergence.

symmetric models with nonlinear pseudosupersymmetry
View Description Hide DescriptionBy applying the higher order Darboux algorithm to an exactly solvable nonHermitian symmetric potential, we obtain a hierarchy of new exactly solvable nonHermitian symmetric potentials with real spectra. It is shown that the symmetry underlying the potentials so generated and the original one is nonlinear pseudosupersymmetry. We also show that this formalism can be used to generate a larger class of new solvable potentials when applied to nonHermitian systems.

Universality of lowenergy scattering in dimensions: The nonsymmetric case
View Description Hide DescriptionFor a very large class of potentials, , we prove the universality of the lowenergy scattering amplitude, . The result is . The only exceptions occur if happens to have a zeroenergy bound state. Our new result includes as a special subclass the case of rotationally symmetric potentials, .

Sz.Nagy–Foias theory and Lax–Phillips type semigroups in the description of quantum mechanical resonances
View Description Hide DescriptionA quantum mechanical version of the Lax–Phillips scattering theory was recently developed. This theory is a natural framework for the description of quantum unstable systems. However, since the spectrum of the generator of evolution in this theory is unbounded from below, the existing framework does not apply to a large class of quantum mechanical scattering problems. It is shown in this work that the fundamental mathematical structure underlying the Lax–Phillips theory, i.e., the Sz.Nagy–Foias theory of contraction operators on Hilbert space, can be used for the construction of a formalism in which models associated with a semibounded spectrum may be accomodated.

Euclidean quantum mechanics in the momentum representation
View Description Hide DescriptionA time reversible probabilistic representation of solutions of the (Euclidean) Schrödinger equation in momentum representation is constructed using Lévy processes and bridges. Each diffusion in the position representation is associated with a jump diffusion in the momentum space. Our method can be looked upon as a rigorous version of Feynman’s path integral approach. Several examples are studied.

Control of finitedimensional quantum systems: Application to a spin particle coupled with a finite quantum harmonic oscillator
View Description Hide DescriptionIn this paper, we consider the problem of the controllability of a finitedimensional quantum system in both the Schrödinger and interaction pictures. Introducing a Quantum Transfer Graph, we elucidate the role of Lie algebra rank conditions and the complex nature of the control matrices. We analyze the example of a sequentially coupled level system: a spin particle coupled to a finite quantum harmonic oscillator. This models an important physical paradigm of quantum computers—the trapped ion. We describe the control of the finite model obtained, under the right conditions, from the original infinitedimensional system.

Asymptotic of complex hyperbolic geometry and spectral analysis of Landaulike Hamiltonians
View Description Hide DescriptionIn this paper we show that the flat Hermitian complex geometry of , , is approximated by the complex hyperbolic geometry of the Bergman complex balls of radius . Furthermore, it will be shown that some elements of the spectral analysis, such as the spectrum, the eigenprojector and the resolvent kernels, associated to the socalled Landaulike Hamiltonian on give rise to their analogous of the Landaulike Hamiltonian on by letting tend to infinity.

Pure point spectrum for the time evolution of a periodically rank kicked Hamiltonian
View Description Hide DescriptionWe find the conditions under which the spectrum of the unitary time evolution operator for a periodically rank kicked system remains pure point. This stability result allows one to analyze the onset of, or lack of chaos in this class of quantum mechanical systems, extending the results for rank1 systems produced by Combescure and others. This work includes a number of unitary theorems equivalent to those well known and used in the selfadjoint theory.

Exact solution to a supersymmetric Gaudin model
View Description Hide DescriptionWe investigate a special case of the Gaudin model related to the superalgebra . We present an exact solution to that system diagonalizing a complete set of commuting observables, and providing the corresponding eigenvectors and eigenvalues. The approach used in this paper is based on the coalgebra symmetry of the model, already known from the Calogero–Gaudin system.

Two interacting electrons in a uniform magnetic field and a parabolic potential: The general closedform solution
View Description Hide DescriptionWe present an analytical analysis of the twodimensional Schrödinger equation for two interacting electrons subjected to a homogeneous magnetic field and confined by a twodimensional external parabolic potential. We have found the general closedform expression for the eigenstates of the problem and its corresponding eigenenergies for particular values of magnetic field and spatial confinement length. The mathematical framework is just based on a rigorous solution of the threeterm recursion relation among the coefficients that arises from the series solution of biconfluent Heun (BHE) equation, connected with the radial part of the Schrödinger equation for the internal motion. It is also shown that, by vanishing of Coulomb repulsion strength, the obtained explicit analytical solutions of BHE equation reduces to the wellknown polynomials satisfying the associated Laguerre differential equation. Furthermore, in the presence of this interaction, the results are compared with those previously obtained in the literature for first few lowlying states, and are found to be in an exact agreement with them.

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


Yang–Mills action from minimally coupled bosons on and on the fourdimensional Moyal plane
View Description Hide DescriptionWe consider bosons on (Euclidean) that are minimally coupled to an external Yang–Mills field. We compute the logarithmically divergent part of the cutoff regularized quantum effective action of this system. We confirm the known result that this term is proportional to the Yang–Mills action. We use pseudodifferential operator methods throughout to prepare the ground for a generalization of our calculation to the noncommutative fourdimensional Moyal plane . We also include a detailed comparison of our cutoff regularization to heat kernel techniques. In the case of the noncommutative space, we complement the usual technique of asymptotic expansion in the momentum variable with operator theoretic arguments in order to keep separated quantum from noncommutativity effects. We show that the result from the commutative space still holds if one replaces all pointwise products by the noncommutative Moyal product.

Variational twofermion wave equations in quantum electrodynamics: Muoniumlike systems
View Description Hide DescriptionWe consider a reformulation of quantum electrodynamics in which covariant Green functions are used to solve for the electromagnetic field in terms of the fermion fields. The resulting modified Hamiltonian contains the photon propagator directly. A simple Fockstate variational trial function is used to derive relativistic twofermion equations variationally from the expectation value of the Hamiltonian of the field theory. The interaction kernel of the equation is shown to be, in essence, the invariant matrix in lowest order. Solutions of the twobody equations are presented for muoniumlike systems for small coupling strengths. The results compare well with the observed muonium spectrum, as well as that for hydrogen and muonic hydrogen. Anomalous magnetic momenteffects are discussed.

Charge superselection sectors for QCD on the lattice
View Description Hide DescriptionWe study quantum chromodynamics(QCD) on a finite lattice in the Hamiltonian approach. First, we present the field algebra as comprising a gluonic part, with basic building block being the crossed product algebra , and a fermionic (CARalgebra) part generated by the quark fields. By classical arguments, has a unique (up to unitary equivalence) irreducible representation. Next, the algebra of internal observables is defined as the algebra of gauge invariant fields, satisfying the Gauss law. In order to take into account correlations of field degrees of freedom inside with the “rest of the world,” we must extend by tensorizing with the algebra of external gauge invariant operators. This way we construct the full observable algebra. It is proved that its irreducible representations are labelled by valued boundary flux distributions. Then, it is shown that there exist unitary operators (charge carrying fields), which intertwine between irreducible sectors leading to a classification of irreducible representations in terms of the valued global boundary flux. By the global Gauss law, these three inequivalent charge superselection sectors can be labeled in terms of the global color charge (triality) carried by quark fields. Finally, is discussed in terms of generators and relations.

Squareintegrable wave packets from the Volkov solutions
View Description Hide DescriptionRigorous mathematical proofs of some properties of the Volkov solutions are presented, which describe the motion of a relativistic charged Dirac particle in a classical, plane electromagnetic wave. The Volkov solutions are first rewritten in a convenient form, which clearly reveals some of the symmetries of the underlying Dirac equation. Assuming continuity and boundedness of the electromagnetic vector potential, it is shown how one may construct squareintegrable wave packets from momentum distributions in the space . If, in addition, the vector potential is and the derivative is bounded, these wave packets decay in space faster than any polynomial and fulfill the Dirac equation. The mapping which takes momentum distributions into wave packets is shown to be isometric with respect to the norm and may therefore be continuously extended to a mapping from . For a momentum function in , an integral representation of this extension is presented.

Upper limit on the critical strength of central potentials in relativistic quantum mechanics
View Description Hide DescriptionIn the context of relativistic quantum mechanics, where the Schrödinger equation is replaced by the spinless Salpeter equation, we show how to construct a large class of upper limits on the critical value, , of the coupling constant, , of the central potential, . This critical value is the value of for which a first wave bound state appears.

Matrix theory of unoriented membranes and Jordan algebras
View Description Hide DescriptionWe express the equations of motion of unoriented membranes with the help of Jordan algebras.

 GENERAL RELATIVITY AND GRAVITATION


Consistent and mimetic discretizations in general relativity
View Description Hide DescriptionA discretization of a continuum theory with constraints or conserved quantities is called mimetic if it mirrors the conserved laws or constraints of the continuum theory at the discrete level. Such discretizations have been found useful in continuum mechanics and in electromagnetism. We have recently introduced a new technique for discretizing constrained theories. The technique yields discretizations that are consistent, in the sense that the constraints and evolution equations can be solved simultaneously, but it cannot be considered mimetic since it achieves consistency by determining the Lagrange multipliers. In this paper we would like to show that when applied to general relativity linearized around a Minkowski background the technique yields a discretization that is mimetic in the traditional sense of the word. We show this using the traditional metric variables and also the Ashtekar new variables, but in the latter case we restrict ourselves to the Euclidean case. We also argue that there appear to exist conceptual difficulties to the construction of a mimetic formulation of the full Einstein equations, and suggest that the new discretization scheme can provide an alternative that is nevertheless close in spirit to the traditional mimetic formulations.

General relativity via complete integrals of the Hamilton–Jacobi equation
View Description Hide DescriptionThe aim of this work is to present a formulation to general relativity, which is analogous to the null surface formulation, but now instead of starting with a complete integral of the eikonal equation we start with a complete integral of the Hamilton–Jacobi equation. In the first part of this work we show that on the space of solutions of a certain class of systems of six secondorder partial differential equations,, , , a fourdimensional (definite or indefinite) metric, , can be constructed on the fourdimensional solution space with local coordinates . Furthermore the solutions,, satisfy the fourdimensional Hamilton–Jacobi equation,. We remark that this structure is invariant under a subset of contact transformations. In the next section, as an example, we apply these results to the Schwarzschild metric. Finally we use the fourdimensional metric obtained in the first part and we impose the Einstein equations.

 DYNAMICAL SYSTEMS


An algebrogeometric solution for a Hamiltonian system with application to dispersive long wave equation
View Description Hide DescriptionBy using an iterative algebraic method, we derive from a spectral problem a hierarchy of nonlinear evolution equations associated with dispersive long wave equation. It is shown that the hierarchy is integrable in Liouville sense and possesses biHamiltonian structure. Two commutators, with zero curvature and Lax representations, for the hierarchy are constructed, respectively, by using two different systematic methods. Under a Bargmann constraint the spectral is nonlinearized to a completely integrable finite dimensional Hamiltonian system. By introducing the Abel–Jacobi coordinates, an algebrogeometric solution for the dispersive long wave equation is derived by resorting to the Riemann theta function.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Lagrange–Fedosov nonholonomic manifolds
View Description Hide DescriptionWe outline a unified approach to geometrization of Lagrange mechanics, Finsler geometry and geometric methods of constructing exact solutions with generic offdiagonal terms and nonholonomic variables in gravity theories. Such geometries with induced almost symplectic structure are modeled on nonholonomic manifolds provided with nonintegrable distributions defining nonlinear connections. We introduce the concept of Lagrange–Fedosov spaces and Fedosov nonholonomic manifolds provided with almost symplectic connection adapted to the nonlinear connection structure. We investigate the main properties of generalized Fedosov nonholonomic manifolds and analyze exact solutions defining almost symplectic Einstein spaces.
