Volume 36, Issue 8, August 1995
Index of content:

Bender–Wu formula for the Zeeman effect
View Description Hide DescriptionWe prove the Bender–Wu formula in the case of magnetic Schrödinger operators under general considerations on the potential. We also give a bound on the behavior of the perturbation series coefficients for the ground state energy.

Generalized fermions and Lie superalgebras spl(m/n)
View Description Hide DescriptionDifferent statistics generalizing the Fermi–Dirac one, i.e., statistics associated with parafermions of order p, orthofermions, deformed parafermions, and genons, are studied in connection with Lie superalgebras. The class spl(m/n) appears as the fundamental one in this context. A specific relation between parafermions and orthofermions is also pointed out.

Deformation methods for single paraoscillator systems
View Description Hide DescriptionThe origin of and distinction between the two deformation techniques currently in use for single particle paraoscillators is made clear. For paraoscillators deformed using the more familiar method, algebra deformation, (i) a single deformation function unifying parabosons and parafermions is provided and (ii) Fock space representations of all single paraoscillator systems as bilinear combinations of deformed oscillators are obtained. For the (distinct) Fock space deformation method, the deformed paraoscillator algebras are worked out in detail and their Casimir operators are given. Difficulties concerning the relationship between the single particle Calogero–Vasiliev oscillator and the single particle parabose system are resolved.

The geometrical approach to multidimensional inverse scattering
View Description Hide DescriptionWe prove that in multidimensional short‐range potential scattering the high velocity limit of the scattering operator of an N‐body system determines uniquely the potential. For a given long‐range potential the short‐range potential of the N‐body system is uniquely determined by the high velocity limit of the modified Dollard scattering operator. Moreover, we prove that any one of the Dollard scattering operators determines uniquely the total potential. We obtain as well a reconstruction formula with an error term. Our simple proof uses a geometrical time‐dependent method.

Renormalized perturbation series and the semiclassical limit of quantum mechanics
View Description Hide DescriptionThe accuracy of the renormalized perturbation series for anharmonic oscillators and other simple quantum‐mechanical models increases with the quantum number when the adjustable parameter is determined by a simple variational criterion. In the semiclassical limit of quantum mechanics such an expansion converges towards the result of the Jeffreys–Wentzel–Kramers–Brillouin method to all orders. The hypervirial and Hellmann–Feynman theorems facilitate the derivation of the semiclassical series from Rayleigh–Schrödinger perturbation theory.

On the construction of free fields in relativistic quantum mechanics
View Description Hide DescriptionA general discussion of the construction of free fields based on Weinberg anszatz is provided and various applications appearing in the literature are considered.

O(3,3)‐like symmetries of coupled harmonic oscillators
View Description Hide DescriptionIn classical mechanics, the system of two coupled harmonic oscillators is shown to possess the symmetry of the Lorentz groupO(3,3) or SL(4,r) in the four‐dimensional phase space. In quantum mechanics, the symmetry is reduced to that of O(3,2) or Sp(4), which is a subgroup of O(3,3) or SL(4,r), respectively. It is shown that among the six Sp(4)‐like subgroups, only one possesses the symmetry which can be translated into the group of unitary transformations in quantum mechanics.

Structure of the chiral scalar superfield in ten dimensions
View Description Hide DescriptionWe describe the tensors and spinor‐tensors included in the θ‐expansion of the ten‐dimensional chiral scalar superfield. The product decompositions of all the irreducible structures with θ and the θ^{2}tensor are provided as a first step toward the obtention of a full tensorcalculus for the superfield.

An application of the Peter–Weyl theorem to non‐Abelian lattice gauge theory
View Description Hide DescriptionThis article presents an application of the Peter–Weyl theorem to obtain a new formula for the physical states of the Kogut–Susskind Hamiltonian model. In this formula a physical state is identified with a matrix element of a suitable representation of the infinite product of the gauge group. The Clebsch–Gordan coefficients for the physical states are obtained. A handy formula is deduced for the plaquette term in the Hamiltonian, so that the matrix elements of the Hamiltonian in the physical space for the (2+1)‐dimensional SU(2) model can be calculated without using the Wigner–Eckart theorem for tensor operators.

Multidimensional extension of a Wentzel–Kramers–Brillouin improvement for spherical quantum billiard zeta functions
View Description Hide DescriptionSome insight is offered into the dimensional dependence of the Wentzel–Kramers–Brillouin (WKB) and improved‐WKB approximations introduced by Steiner in connection with hyperspherical (originally circular) quantum billiards. The accuracy of every new D‐dimensional picture is estimated by examining the residues of the spectral zeta‐function poles.

Operator analysis of nonrenormalizable multicomponent ultralocal field models
View Description Hide DescriptionA nontrivial, nonperturbative quantum field theory of O(N)‐invariant multicomponent, nonrenormalizable ultralocal models is developed along lines presented earlier for single‐component models. The additional, nonclassical, repulsive potential that is always present in the solution of the single‐component case becomes indefinite, and may even vanish in the multicomponent case. The disappearance of the nonclassical and singular potential does not mean a return to a conventional quantum field theory, and the operator solution of multicomponent ultralocal fields remains noncanonical. Nontrivial, i.e., non‐Gaussian, results hold for any number N of components, and suitable nontrivial behavior persists, even in the infinite‐component (N=∞) case as well.

Nontrivial path integrals for nonrenormalizable fields—Multicomponent ultralocal models
View Description Hide DescriptionA nontrivial lattice‐space path integral formulation of nonrenormalizable multicomponent ultralocal models is constructed from the nonperturbative operator solutions presented in a recent paper. The indefinite, nonclassical, singular potential required for the nontriviality has different effects on distributions compared to the single‐component case, however, the essential property of reweighting the distribution at the origin is similar. The appearance of additional nonclassical, singular potentials suggests that we cannot always place the classical Lagrangian or classical Hamiltonian directly into the path‐integral formulation, or in other words, a straightforward canonical quantization of fields with infinite degrees of freedom does not always apply.

An optimization approach to a three‐dimensional acoustic inverse problem in the time domain
View Description Hide DescriptionAn optimization approach to a three‐dimensional acoustic inverse problem is considered in the time‐domain. The velocity and the density are reconstructed by minimizing an objective functional. By introducing a dual function, the gradient of the objective functional is found with an explicit expression. The parameters are then reconstructed by an iterative algorithm (the conjugate gradient method). The uniqueness of the solution is also proved.

Long‐range effects in asymptotic fields and angular momentum of classical field electrodynamics
View Description Hide DescriptionAsymptotic properties of classical field electrodynamics are considered. Special attention is paid to the long‐range structure of the electromagnetic field. It is shown that conserved Poincaré quantities may be expressed in terms of the asymptotic fields. Long‐range variables are shown to be responsible for an angular momentum contribution which mixes Coulomb and infrared free field characteristics; otherwise angular momentum and energy‐momentum separate into electromagnetic and matter fields contributions.

A new method of solution of Hallén’s problem
View Description Hide DescriptionA wide class of electromagnetic scattering problems can be expressed as a system of dual integral equations. This kind of integral equation, occurring in boundary value problems wherein there is one equation for a certain region and another for the dual domain, is usual in diffraction problems. Considerable attention has been drawn by many researchers in the field of optics, acoustics, scattering of elastic waves, accelerator physics, and antennatheory. Wiener–Hopf techniques enable us to solve such kinds of integral equations when the two regions are contiguous and semi‐infinite. Unfortunately Wiener–Hopf techniques do not apply if one of the regions is finite; this is the case for realistic scatterers, such as irises, antennas, and drift tubes in accelerators. In this article a general method for solving such dual integral equations is discussed and applied to a particular case: Hallén’s equation of cylindrical antennas. This method consists of a transformation of the dual integral equations into a Fredholm integral equation of the second kind and then into a linear system of algebraic equations. Comparisons with results obtained by other methods for a wide range of frequencies show the accuracy and the robustness of the proposed one.

Multivariable Grassmann integrals and the high temperature expansion of interacting fermionic models
View Description Hide DescriptionUsing the noncommuting nature of fermionic fields we obtain a nonperturbative method to calculate the high temperature limit of the grand canonical partition function of interacting fermionic models. This nonperturbative approach is applied to the Hubbard model in d=2(1+1) and we recover the known results in the limit T → ∞.

Entropy of orthogonal polynomials with Freud weights and information entropies of the harmonic oscillator potential
View Description Hide DescriptionThe information entropy of the harmonic oscillator potential V(x)=1/2λx ^{2} in both position and momentum spaces can be expressed in terms of the so‐called ‘‘entropy of Hermite polynomials,’’ i.e., the quantity S _{ n }(H):= −∫_{−∞} ^{+∞} H ^{2} _{ n }(x)log H ^{2} _{ n }(x) e ^{−x 2 } dx. These polynomials are instances of the polynomials orthogonal with respect to the Freud weights w(x)=exp(−‖x‖^{ m }), m≳0. Here, a very precise and general result of the entropy of Freud polynomials recently established by Aptekarev et al. [J. Math. Phys. 35, 4423–4428 (1994)], specialized to the Hermite kernel (case m=2), leads to an important refined asymptotic expression for the information entropies of very excited states (i.e., for large n) in both position and momentum spaces, to be denoted by S _{ρ} and S _{γ}, respectively. Briefly, it is shown that, for large values of n, S _{ρ}+1/2logλ≂log(π√2n/e)+o(1) and S _{γ}−1/2log λ≂log(π√2n/e)+o(1), so that S _{ρ}+S _{γ}≂log(2π^{2} n/e ^{2})+o(1) in agreement with the generalized indetermination relation of Byalinicki‐Birula and Mycielski [Commun. Math. Phys. 44, 129–132 (1975)]. Finally, the rate of convergence of these two information entropies is numerically analyzed. In addition, using a Rakhmanov result, we describe a totally new proof of the leading term of the entropy of Freud polynomials which, naturally, is just a weak version of the aforementioned general result.

Global smooth solution for the Klein–Gordon–Zakharov equations
View Description Hide DescriptionIn this paper the initial value problem to the Klein–Gordon–Zakharov equations in two dimensions is discussed. Without assuming that the Cauchy data are small, we prove the existence and uniqueness of the global smooth solution for the problem via the so‐called continuous method and delicate a priori estimates. Also the asymptotic behaviour of the solution to the K–G–Z equations with a small parameter approaching zero is studied.

A class of exact, periodic solutions of nonlinear envelope equations
View Description Hide DescriptionA class of periodic solutions of nonlinear envelope equations, e.g., the nonlinear Schrödinger equation (NLS), is expressed in terms of rational functions of elliptic functions. The Hirota bilinear transformation and theta functions are used to extend and generalize this class of solutions first reported for NLS earlier in the literature. In particular a higher order NLS and the Davey–Stewartson (DS) equations are treated. Doubly periodic standing waves solutions are obtained for both the DSI and DSII equations. A symbolic manipulation software is used to confirm the validity of the solutions independently.

Symmetries and constants of the motion for higher‐order Lagrangian systems
View Description Hide DescriptionWe obtain a classification of infinitesimal symmetries of an autonomous higher‐order Lagrangian system by using the theory of lifts of vector fields to tangent bundles. Also, a classification of infinitesimal symmetries of a time‐dependent higher‐order Lagrangian system is obtained. The relationship between infinitesimal symmetries and constants of the motion is investigated.