Volume 25, Issue 9, September 1984
Index of content:

Tensor operator realizations of E _{6} and structural zeros of the 6j‐symbol
View Description Hide DescriptionFor the exceptional Lie algebraE _{6} minimal SO(3) tensor operator realizations are constructed which correspond to the maximal decomposition of E _{6} in the respective chains E _{6}⊃F _{4}⊃SO(3), E _{6}⊃SU(3)⊗G _{2}⊃SO(3)⊗SO(3), E _{6}⊃F _{4}⊃SO(3)⊗G _{2}⊃SO(3)⊗SO(3), E _{6}⊃Sp(8)⊃SO(3), and E _{6}⊃G _{2}⊃SO(3). Two particular realizations are shown to provide a basis in which certain structural zeros of Racah’s 6j‐symbol can be explained.

On some general properties of the point spectrum of three particles moving in one dimension
View Description Hide DescriptionThe eigenstates of three particles moving in one dimension are classified according to the S _{3} plus parity group. The ordering of the ground state S _{3} band is given for a fairly general class of potentials. Sufficient conditions are given both for existence and nonexistence of bound states of a given symmetry.

On the classical part of the mean field dynamics for quantum lattice systems in grand canonical representations
View Description Hide DescriptionFor a class of discrete mean field models the limiting dynamics is investigated in the representations of generalized grand canonical states. It is demonstrated that for a certain form of spontaneous symmetry breakdown the W*‐automorphism dynamics exhibits a uniquely determined nontrivial classical part, which is essential for the explanation of macroscopic quantum phenomena.

Group theoretic treatment of the Dirac–Coulomb equation and matrix elements of its tensor operators
View Description Hide DescriptionThe solution of the Dirac–Coulomb equation obtained by Wong and Yeh is interpreted in terms of the SU(2)×SO(2,1) group. All electromagnetic transition probabilities can be considered as matrix elements of the tensor operators of this group, and evaluated exactly. The cases considered include transitions from bound‐to‐bound, bound‐to‐continuum, and continuum‐to‐continuum states.

The direct linearization of a class of nonlinear evolution equations
View Description Hide DescriptionThe paper deals with the direct linearization, an approach used to generate particular solutions of the partial differential equations that can be solved through the inverse scatteringtransform. Linear integral equations are presented which enable one to find broad classes of solutions to certain nonlinear evolution equations in 1+1 and 2+1 dimensions.

On the old–new method of solving nonlinear equations
View Description Hide DescriptionA method for solving a quasilinear nonelliptical equation of the second order is developed. We give classification and parametrization of simple elements of the equation. An equation of potential stationary flow of compressible gas in a supersonic region is considered as an example. A new exact solution is obtained which may be treated as a nonlinear analog of stationary wave.A gauge structure for the equation and an analog of Bäcklund transformation are introduced.

V*‐algebras: A particular class of unbounded operator algebras
View Description Hide DescriptionWe consider a weak unbounded commutant for a set of unbounded operators and we examine op*‐algebras which coincide with some of their bicommutant. This class of op*‐algebras, called V*‐algebras, shows some properties close to those which hold true for bounded operator algebras.

On the center‐of‐mass motion of geometrically confined classical particles
View Description Hide DescriptionBy means of group‐theoretical methods based on O(3,2) a description of center‐of‐mass motion is given of a set of harmonically oscillating classical particles which can attain relativistic velocities. The limitation ‖v‖<c on the velocity leads to a limitation ‖r‖<R on the amplitude, where R is related to the universal oscillator frequency ω by R=cω^{−} ^{1}. It turns out that the center‐of‐mass carries out a harmonic oscillation with the same frequency ω and the same limitations, and that conditions can be formulated for the set of particles to be in its ‘‘rest system.’’ The method can be applied to hadrons considered as bags containing harmonically oscillating classical quarks.

Homogeneous canonical formulation of the nonrelativistic hydrogen atom
View Description Hide DescriptionThe homogeneous canonical formulation is applied to the Hamiltonian of the nonrelativistic hydrogen atom. Its connection with the isotropic harmonic oscillator in a four‐dimensional Riemann space leads to the quantum analog of the Kepler problem.

Double phase‐integral approximations: A systematic simplification technique for wave equations with cutoffs and resonances
View Description Hide DescriptionThe time‐independent wave equation,d ^{2}ψ/d z ^{2}+Q ^{2}(z)ψ=0, where Q ^{2}(z) may have arbitrary order zeros and poles on or close to the real axis, is transformed to a simpler wave equation of similar properties (model). Approximate transformations leading from the original wave equation to the model are simply related to Fröman’s higher‐order phase integrals, but are nevertheless well defined at the pertinent zeros and poles of Q ^{2}(z).

Transmission through cutoffs and resonances in the double phase‐integral approximation
View Description Hide DescriptionUsing the double phase‐integral approximation technique developed earlier for the wave equationd ^{2}ψ/d z ^{2}+Q ^{2}(z) ψ=0, we derive analytical formulas for the reflection (R), transmission (T), and absorption (A) coefficients. They are valid to arbitrary order in the expansion parameter, for functions Q ^{2}(z) having either two cutoffs or one cutoff and one resonance. For two examples of this type the formulas for R, T, and A are checked against numerical results, using approximations up to fifth order.

Coherent state theory of the noncompact symplectic group
View Description Hide DescriptionAn extended coherent statetheory is presented for the noncompact Sp(3,R) group which reveals a simple relationship between the Sp(3,R) algebra and its contracted u(3)‐boson limit. The relationship is used to derive a remarkably accurate analytic expression for Sp(3,R) matrix elements for the generic lowest‐weight representations. The expression is shown to be exact whenever the states involved are multiplicity free with respect to the u(3) subalgebra. It is further shown how exact matrix elements are easily calculated in general. Dyson and Holstein–Primakoff type u(3)‐boson expansions are given.

Remarks about inverse diffraction problem
View Description Hide DescriptionFrom the scattering data one finds the support function or the principal curvatures of the surface of a reflecting obstacle. From either of these data the surface is effectively reconstructed.

Cylindrically symmetric solitary wave solutions to the Einstein equations
View Description Hide DescriptionThe integration of the Einstein equations for cylindrically symmetric solitary waves is reduced to a single quadrature when the ‘‘seed’’ solution is diagonal. Also in this case, explicit formulas that show the solitary wave character of the one‐ and two‐soliton solutions are studied. A particular case of n‐soliton solution is exhibited. Two theorems that show how to construct new solutions from known ones are presented.

From i ° to the 3+1 description of spatial infinity
View Description Hide DescriptionBy carrying out a 3+1 decomposition of the spi framework, the expressions of the conserved quantities, defined at i ° in terms of the Weyl curvature, are recast in terms of the initial data of the physical space‐time. In particular, the analysis brings out the supertranslation ambiguities in the usual 3+1 definitions of angular momentum and clarifies, within the 3+1 framework, the meaning of the stronger boundary condition needed to remove these ambiguities. The discussion is so arranged that only a minimal acquaintance with the spi framework is necessary to appreciate these issues.

Finite energy electric monopoles in an extended theory of gravitation
View Description Hide DescriptionWe present a one‐parameter family of extended Einstein–Maxwell Lagrangians in which an antisymmetric tensor field is nonlinearly coupled to both the gravitational and electromagnetic fields. We show that for arbitrary, positive values of the relevant parameter, the theory admits exact, static spherically symmetric solutions with everywhere finite electric field density and energy density. Asymptotically, the solutions are indistinguishable from the Reissner–Nordstrom solution in general relativity. In addition, we show that a corrected form of the exact solution in the nonsymmetric Kaluza–Klein theory presented in an earlier paper provides a special case of the family of solutions described above.

Massless fermions and Kaluza–Klein theory with torsion
View Description Hide DescriptionA pure Kaluza–Klein theory contains no massless fermion in four‐dimensional theory. We investigate the effect of introducing torsion on the internal manifold and find that there are massless fermions. The hope is that given an isometry group the representation to which these fermions belong is fixed, in contrast to the situation in Yang–Mills theory. We show that this is indeed the case, but the representations do not appear to be the ones favored by current theoretical prejudice. The cases with parallelizable torsions on a group manifold as the internal manifold are analyzed in detail.

An infinite chain of inequalities for correlation functions of classical lattice systems
View Description Hide DescriptionIn quantum‐mechanical systems the moments of the symmetrized time‐dependent autocorrelation function satisfy an infinite set of chained inequalities. By analogy the same inequalities are derived for correlation functions of classical lattice systems.

Kinetic potentials in quantum mechanics
View Description Hide DescriptionSuppose that the Hamiltonian H=−Δ+v f(r) represents the energy of a particle which moves in an attractive central potential and obeys nonrelativistic quantum mechanics. The discrete eigenvaluesE _{ n l }=F _{ n l }(v) of H may be expressed as a Legendre transformation F _{ n l }(v)=min_{ s≳0} ( s+v f̄_{ n l }(s)), n=1,2,3,..., l=0,1,2,..., where the ‘‘kinetic potentials’’ f̄_{ n l }(s) associated with f(r) are defined by f̄_{ n l }(s) =inf_{ D n l } sup_{ψ∈D n l , ∥ψ∥=1} ∫ ψ(r) f ([ψ,−Δψ)/s]^{1} ^{/} ^{2} r)ψ(r)d ^{3} r, and D _{ n l } is an n‐dimensional subspace of L ^{2}(R^{3}) labeled by Y _{ l } ^{ m }(θ,φ), m=0, and contained in the domain D(H) of H. If the potential has the form f(r)=∑^{ N } _{ i=1} g ^{(i)} ( f ^{(i)}(r)) then in many interesting cases it turns out that the corresponding kinetic potentials can be closely approximated by ∑^{ N } _{ i=1} g ^{(i)} ( f̄_{ n l } ^{(i)}(s)). This nice behavior of the kinetic potentials leads to a constructive global approximation theory for Schrödinger eigenvalues. As an illustration, detailed recipes are provided for arbitrary linear combinations of power‐law potentials and the log potential. For the linear plus Coulomb potential and the quartic anharmonic oscillator the approximate eigenvalues are compared to accurate values found by numerical integration.

Sensitivity analysis of stochastic kinetic models
View Description Hide DescriptionA formalism for sensitivity analysis of stochastic models describing fluctuation phenomena in chemically reacting systems is developed. The method is not restricted to chemical kinetics and can be used to analyze any model of a physical system whose state variables obey stochastic differential equations with white noise. Expressions for the sensitivity coefficients and densities are obtained. These expressions are suitable for direct evaluation by means of a stochastic simulation in a computer. The relationship between these quantities and the response functions studied in statistical mechanics is discussed.