Volume 38, Issue 10, October 1997
Index of content:

The quartic anharmonic oscillator and its associated nonconstant magnetic field
View Description Hide DescriptionQuantum mechanical anharmonic oscillators and Hamiltonians for particles in external magnetic fields are related to representations of nilpotent groups. Using this connection the eigenfunctions of the quartic anharmonic oscillator with potential can be used to determine the eigenfunctions of a charged particle in a nonconstant magnetic field, of the form The quartic anharmonic oscillatoreigenvalues for lowlying states are obtained numerically and a function which interpolates between (a double harmonic oscillator) and (a harmonic oscillator) is shown to give a good fit to the numerical data. Approximate expressions for the quartic anharmonic oscillator eigenfunctions are then used to get the eigenfunctions for the magnetic field Hamiltonian.

A sufficient condition for the existence of bound states in a potential
View Description Hide DescriptionFor a wide class of purely attractive potentials, we obtain a new sufficient condition for the existence of bound states for any angular momentum. Applied to some exactly soluble cases, the condition gives good results as compared to exact results.

Bounds on Schrödinger eigenvalues for polynomial potentials in dimensions
View Description Hide DescriptionIf a single particle obeys nonrelativistic in and has the Hamiltonian then the lowest eigenvalue is given approximately by the semiclassical expression It is proved that this formula yields a lower bound when and an upper bound when An extension is made to allow for a Coulomb term when The general formula is applied to the examples and in dimensions 1 to 10, and the results are compared to accurate eigenvalues obtained numerically.

Hypercomplex numbers and the description of spin states
View Description Hide DescriptionA family of hypercomplex numbers is introduced in which multiplication is commutative and members can have up to eight components. In particular, the eight basis elements contain those for ordinary complex numbers, , as well as new elements where ; the operation * being the generalization of complex conjugation. This family lends itself to the description of quantum mechanical spin states in that it offers a simple treatment of time reversal, representations with the same conjugation properties as underlying operators, and explicit continuousangle spherical harmonic functions analogous to the for orbital angular momentum. The new elements are especially well suited for halfintegral spin states, whereas conventional complex numbers remain useful for integral spin states.

On the universal Chamseddine–Connes action. I. Details of the action computation
View Description Hide DescriptionWe give a detailed computation of the bosonic action of the Chamseddine–Connes model which we performed using different techniques.

Gauge transformations in relativistic twoparticle constraint theory
View Description Hide DescriptionUsing connection with quantum field theory, the infinitesimal covariant Abelian gauge transformation laws of relativistic twoparticle constraint theorywave functions and potentials are established and weak invariance of the corresponding wave equations shown. Because of the threedimensional projection operation, these transformation laws are interaction dependent. Simplifications occur for local potentials, which result, in each formal order of perturbation theory, from the infrared leading effects of multiphoton exchange diagrams. In this case, the finite gauge transformation can explicitly be represented, with a suitable approximation and up to a multiplicative factor, by a momentum dependent unitary operator that acts in space as a local dilatation operator. The latter is utilized to reconstruct from the Feynman gauge the potentials in other linear covariant gauges. The resulting effective potential of the final Pauli–Schrödinger type eigenvalue equation has the gauge invariant attractive singularity leading to a gauge invariant critical coupling constant

Thermal field dynamics and bialgebras
View Description Hide DescriptionIn thermal field dynamics, thermal states are obtained from restrictions of vacuum states on a doubled field algebra. It is shown that the suitably doubled Fock representations of the Heisenberg algebra do not need to be introduced by hand but can be canonically handed down from deformations of the extended Heisenberg bialgebra. No artificial redefinitions of fields are necessary to obtain the thermal representations and the case of arbitrary dimension is considered from the beginning. Our results support a possibly fundamental role of bialgebra structures in defining a general framework for thermal field dynamics.

Generating functionals of physical vertex operators in superstring
View Description Hide DescriptionWe define operators as operators that are already normally ordered with respect to the Friedan–Martinec–Shenker spinor operator. With the help of thus defined operators, we can give the generating functional of physical vertex operators (GFPVO) of fermionic particles (i.e., in the Ramond sector of the superstring). We also propose GFPVO of bosonic particles (i.e., in the Neveu–Schwarz sector of the superstring), which is simply obtained by supergeneralizing GFPVO in the bosonic string.

Coherent states map for MIC–Kepler system
View Description Hide DescriptionThe coherent states map for MIC–Kepler system is constructed. The quantization of this system is given by the coherent states method.

Logarithmic potential of Hermite polynomials and information entropies of the harmonic oscillator eigenstates
View Description Hide DescriptionThe problem of calculating the information entropy in both position and momentum spaces for the th stationary state of the onedimensional quantum harmonic oscillator reduces to the evaluation of the logarithmic potential at the zeros of the Hermite polynomial Here, a closed analytical expression for is obtained, which in turn yields an exact analytical expression for the entropies when the exact location of the zeros of is known. An inequality for the values of at the zeros of is conjectured, which leads to a new, nonvariational, upper bound for the entropies. Finally, the exact formula for is written in an alternative way, which allows the entropies to be expressed in terms of the evenorder spectral moments of the Hermite polynomials. The asymptotic limit of this alternative expression for the entropies is discussed, and the conjectured upper bound for the entropies is proved to be asymptotically valid.

On the causal structure of Minkowski spacetime
View Description Hide DescriptionThe causal structure of Minkowski spacetime is discussed, in terms of the notions of causal complementation and causal completion. These geometric notions are relevant for quantum field theory and the theory of the KleinGordon equation. Particular attention is given to closed, convex, causally complete subsets of , and the properties of such sets are discussed. The study of such sets is motivated by potential applications to the theory of local nets of von Neumann algebras. The notion of the envelope of uniqueness of a subset of , familiar from the theory of the wave equation, is discussed, and some results about the relation of this envelope to the causal completion of the set are presented.

oscillators with arbitrary and perturbation expansions with Sturmians
View Description Hide DescriptionIn contrast to widespread belief the current RayleighSchrödinger perturbation theory may provide an easy description of double well oscillators and/or of the strongly anharmonic forces with an arbitrary powerlaw asymptotical growth. One has only to work in a suitable Sturmian basis. The feasibility and numerical efficiency of the construction is illustrated on a few onedimensional onebody examples.

Mechanical systems with nonholonomic constraints
View Description Hide DescriptionA geometric setting for the theory of firstorder mechanical systems subject to general nonholonomic constraints is presented. Mechanical systems under consideration are not supposed to be Lagrangian systems, and the constraints are not supposed to be of a special form in the velocities (as, e.g., affine or linear). A mechanical system is characterized by a certain equivalence class of 2forms on the first jet prolongation of a fibered manifold. The nonholonomic constraints are defined to be a submanifold of the first jet prolongation. It is shown that this submanifold is canonically endowed with a distribution—this distribution (resp., its vertical subdistribution) has the meaning of generalized possible (resp., virtual) displacements. The concept of a constraint force is defined, and a geometric version of the principle of virtual work is proposed. From the principle of virtual work a formula for a workless constraint force is obtained. A mechanical system subject nonholonomic constraints is modeled as a deformation of the original (unconstrained) system. A direct characterization of a constrained system by means of a class of 2forms along the canonical distribution is given, and “constrained equations of motion” in an intrinsic form are found. A geometric definition of regularity for systems under nonholonomic constraints is provided. In particular, the case of Lagrangian systems is discussed. Also systems subject to holonomic constraints and nonholonomic constraints affine in the velocities are investigated within the range of the general scheme.

Casimir free energy of a spherical cavity in a dielectric medium
View Description Hide DescriptionAn expression is derived for the Casimir free energy of a spherical cavity in a polar dielectric medium at finite temperature. In the process of the derivation the general problem of infinities and their renormalization in calculations of Casimir forces is analyzed. It is shown that the renormalized Casimir free energy has a minimum at a finite mesoscopic value of the cavity radius with the repulsion for and the attraction for The implications of this result for the explanation of cavitation effects are discussed.

A model of continuous polymers with random charges
View Description Hide DescriptionWe study a model of polymers with random charges; the possible shapes of the polymer are represented by the sample paths of a Brownian motion, and the cumulative charge distribution along a polymer is modeled by a realization of a Brownian bridge. Charges interact through a general positivedefinite twobody potential. We study the infinite volume free energy density for a fixed realization of the Brownian motion; this is not selfaveraging, but shows on the contrary a sample dependence through the local time of the Brownian motion. We obtain an explicit series representation for the free energy density; this has a finite radius of convergence, but the free energy is nevertheless analytic in the inverse temperature in the physical domain.

On the second law of thermodynamics for a dissipative system
View Description Hide DescriptionIn this paper we consider dissipative fluxes as well as the conserved variables as a set of independent variables characterizing the thermodynamic state of a nonequilibrium system. We then generalize the traditional internal energy balance equation so that contributions due to the dissipative fluxes are taken into consideration. On the other hand, the second law is formulated in terms of Caratheodory’s inaccessibility condition in conjunction with the assumption that dissipative energy associated with internal work arising from irreversible processes be semipositive definite. We show that the second law formulated in this manner is equivalent to Kelvin’s principle and Clausius’s principle, as well as Clausius’s inequality.

Adler–Kostant–Symes construction, biHamiltonian manifolds, and KdV equations
View Description Hide DescriptionThis paper focuses a relation between Adler–Kostant–Symes (AKS) theory applied to Fordy–Kulish scheme and biHamiltonian manifolds. The spirit of this paper is closely related to Casati–Magri–Pedroni work on Hamiltonian formulation of the KP equation. Here the KdV equation is deduced via the superposition of the Fordy–Kulish scheme and AKS construction on the underlying current algebra This method is different from the Drinfeld–Sokolov reduction method. It is known that AKS construction is endowed with biHamiltonian structure. In this paper we show that if one applies the Fordy–Kulish construction in the Adler–Kostant–Symes scheme to construct an integrable equation associated with symmetric spaces then this superposition method becomes closer to Casati–Magri–Pedroni’s biHamiltonian method of the KP equation. We also add a selfcontained Appendix, where we establish a direct relation between AKS scheme and biHamiltonian methods.

Hierarchies of evolution equations associated with certain symmetric spaces
View Description Hide DescriptionHierarchies of evolution equations associated with certain symmetric spaces are presented. Main examples of such symmetric spaces are Hermitian symmetric spaces and the obtained evolution equations include generalized nonlinear Schrödinger equations. Those results are obtained by extending the Lie algebra in the Ablowitz–Kaup–Newell–Segur scheme from sl(2,C) to more general one.

On solutions of constrained Kadomtsev–Petviashvili equations: Grammians
View Description Hide DescriptionWe show the existence of Grammiantype solutions for the (vector) constrained Kadomtsev–Petviashvili (KP) equations. To introduce the method we give a novel proof for the presence of Grammian solutions for the bilinear modified KP hierarchies.

Initial value problem of the linearized Benjamin–Ono equation and its applications
View Description Hide DescriptionWe consider the initial value problem of the Benjamin–Ono (BO) equation linearized about the soliton solution. By establishing the completeness relation for the eigenfunctions of the linearized BO equation, we construct the explicit solution to this problem. As an application of the above result, we investigate the linear stability of the soliton solution. We show that the wave under consideration is stable against infinitesimal perturbations. Thus we have a direct multisoliton perturbation theory for the BO equation without recourse to the inverse scattering transform. In particular, we can handle the firstorder solution beyond the adiabatic approximation. The completeness relation established here enables us to give a general scheme for evaluating the firstorder correction to the leadingorder soliton solution. We also demonstrate that the firstorder solution satisfies an infinite set of conservation laws modified by the perturbations. Finally, in the onesoliton case, we perform explicit calculations of the firstorder corrections for two different dissipative perturbations that arise in real physical systems and analyze their large time asymptotics.