Volume 20, Issue 6, June 1979
Index of content:

Variational nature of eigenvalue problems with a parameter
View Description Hide DescriptionThe following theorem is proved. Let D (ω) be an operator with eigenvalues and eigenfunctions {d _{ k }(ω), ν_{ k }(ω) }, where ω is a complex parameter. Given a complex number d _{ k } _{0}, let ω_{0} be such that d _{ k }(ω_{0}) =〈 _{ k }(ω_{0}) ‖D (ω_{0}) ‖ν_{ k }(ω_{0}) 〉=d _{ k } _{0}, where _{ k }(ω_{0}) is the dual eigenfunction to ν_{ k }(ω_{0}). Suppose ψ and approximate ν_{ k }(ω_{0}) and _{ k }(ω_{0}), respectively, to order ε. Then, if D (ω) is analytic in ω in the neighborhood of ω_{0}, and if ω′ is such that 〈‖D (ω′) ‖ψ〉=d _{ k } _{0}, ω′ usually will approximate ω_{0} to order ε^{2}. By applying this theorem it is shown that roots of the inhomogeneous plasma dispersion relation usually will be accurate to second order if the associated normal modes and their duals are known merely to first order. The theorem can also be applied to solutions of the dispersion relation in a truncated function space.

Extension of KMS states and angular momentum
View Description Hide DescriptionWe present a description of β‐KMS equilibrium states with respect to some one‐parameter subgroup of (T×R ^{3}) ⧠K, over an asymptotically Abelian algebra of observables (here T×R ^{3} is the group of time and space translations and K a compact group acting on R ^{3}). In the case where K=SO(3), the group of space rotations, we get a characterization of the angular momentum of such states as a parameter occurring in their description, similar to the chemical potential, and classify all of them according to their invariance group, a closed subgroup of T×R×SO(2).

A note on Miura’s transformation
View Description Hide DescriptionIt has been observed by Miura that the solutions of the modified Korteweg–deVries equation can be mapped into those of the Korteweg–deVries equation. In this note we show that all of the solutions of the former, decaying sufficiently rapidly as ‖x‖→∞, map into a sparse solution set of the KdV equation. We use certain results regarding the second Painlevé transcendent to exhibit this fact.

Some exceptional electrovac type D metrics with cosmological constant
View Description Hide DescriptionWe investigate all type D solutions of the Einstein–Maxwell equations (with cosmological constant) such that the Debever–Penrose vectors are aligned along the two eigenvectors of the electromagnetic field, in the special case when a direct generalization of the Goldberg–Sachs theorem is not possible. A solution is found which admits no Killing vectors. We also present an extension of the Golberg–Sachs theorem valid for type D metrics.

Global properties of systems quantized via bundles
View Description Hide DescriptionTake a smooth manifoldM and a Lie algebra action (g action) ϑ on M as the geometrical arena of a physical system moving on M with momenta given by ϑ. We propose to quantize the system with a Mackey‐like method via the associated vector bundle ξ_{ρ} of a principal bundle ξ= (P,π,M,H) with model dependent structure group H and with G‐action φ on P lifted from ϑ on M. This (quantization) bundle ξ_{ρ} gives the Hilbert spaceH=L ^{2}(ξ_{ρ},ω) of the system as the linear space of sections in ξ_{ρ} being square integrable with respect to a volume form ω on M; the usual position operators are obtained; φ leads to a vector field representation D (φ_{ρ},ϑ) of g in H and hence to momentum operators. So H carries the quantum kinematics. In this quantization the physically important connection between geometrical properties of the system, e.g., quasicomqleteness of ρ and G maximality of φ_{ρ}, and global properties of its quantized kinematics, e.g., skew‐adjointness of the momenta and integrability of D (φ_{ρ},ϑ) can easily be studied. The relation to Nelson’s construction of a skew‐adjoint nonintegrable Lie algebra representation and to Palais’ local G actions is discussed. Finally the results are applied to actions induced by coverings as examples of nonmaximal φ_{ρ} on E _{ρ} lifted from maximal ϑ on M which lead to direct consequences from the corresponding quantum kinematics.

SU(4) Clebsch–Gordan coefficients for the formation of baryonium and exotic baryons
View Description Hide DescriptionWe give the SU(4) Clebsch–Gordan coefficients required for the formation of exotic hadronic states of the form (Q Q ) and (Q Q Q Q).

Invariant imbedding equations in general geometries: Numerical solution in spherical geometry
View Description Hide DescriptionA method for deriving the invariant imbedding equations in general geometries by the transfer matrix method is developed. Time‐independent monoenergetic neutron transport in a nonmultiplying medium is considered. Specifically, invariant imbedding equations are derived for the plane, spherical, and cylindrical geometries. The reflection and transmission functions are evaluated numerically using the discrete ordinates method, for the case of a spherical shell with a perfectly absorbing core. A source of error in the results obtained is pointed out.

SU(2,1) generation of electrovacs from Minkowski space
View Description Hide DescriptionFor every nonnull Killing vector K of any given electrovac, there exists a group of tranformations H_{K} of the gravitational and electromagnetic potentials of Ernst. This is the group which is a nonlinear representation of SU(2,1) and was developed by Kinnersley on the basis of work by Ehlers, Harrison, and Geroch. For every K of Minkowski space (MS), we compute the set H_{K}(MS) of all electrovacs derived from MS by noniterative application of H_{K}; the results include appropriate null tetrads, the connection forms, the conform tensors, and (in the discussion) the group of all motions of every member of every H_{K}(MS). Each conform tensor is type N _{ p p } (plane gravitational wave) or type D, and the principal null vector(s) are also eigenvectors of the Maxwell field. Except for those K which represent infinitesimal rotations about a timelike 2‐surface of MS followed by null translations in that 2‐surface, each K has a corresponding MS Killing vector L such that the G _{2} generated by K and L has nonnull surfaces of transitivity and is invertible. The discussion covers properties of the principal null rays and the Maxwell fields, Killing tensors of the results (one of the N _{ p p } families admits an irreducible Killing tensor of Segre characteristic [(11)(11)]), and the precise conditions under which a Killing vector of an electrovac is also an MS Killing vector. Also, some deductions are made concerning the Petrov class and principal null ray properties of the second generation electrovacs which would result from further applications of SU(2,1). Those points of MS which are possible singularities of electrovacs in H_{K}(MS) are classified. The conditions under which an electrovac in H_{K}(MS) has all of R ^{4} (except for curvature singularities) as its domain are found; in particular, such an extension to R ^{4} exists whenever the one‐parameter group generated by K has no fixed points or whenever one restricts H_{K} to the Ehlers or Harrison transformations.

Note on the stability of the Schwarzschild metric
View Description Hide DescriptionIt is shown that the standard arguments for the stability of the Schwarzschild metric can be made into a rigorous proof that the numerical values of linear perturbations of Schwarzschild must remain uniformly bounded for all time.

Integral bounds for N‐body total cross sections
View Description Hide DescriptionWe study the behavior of the total cross sections in the three‐ and N‐body scattering problem. Working within the framework of the time‐dependent two‐Hilbert space scattering theory, we give a simple derivation of integral bounds for the total cross section for all processes initiated by the collision of two clusters. By combining the optical theorem with a trace identity derived by Jauch, Sinha, and Misra, we find, roughly speaking, that if the local pairwise interaction falls off faster than r ^{−3}, then σ_{tot}(E) must decrease faster than E ^{−1/2} at high energy. This conclusion is unchanged if one introduces a class of well‐behaved three‐body interactions.

An algebraic approach to Coulomb scattering in N dimensions
View Description Hide DescriptionUsing purely algebraic techniques, based on the larger symmetry group of the Kepler problem, the phase shifts and the scattering amplitude for Coulomb scattering in N dimensions are derived.

Group content of the Foldy–Wouthuysen transformation and the nonrelativistic limit for arbitrary spin
View Description Hide DescriptionThe relationship between Foldy–Wouthuysen and Lorentz transformations has been clarified throughout this paper. We propose a generalized FW transformation connecting two particular realizations of the (m,j) representation of the Poincaré group: the covariant realization and a canonical realization acting on relativistic probability amplitudes. Fermions and bosons must be considered separately because the intrinsic parity of the particle–antiparticle systems is (−1)^{2j }. Thus for fermions we can directly take the 2(2j+1) ‐ dimensional Joos–Weinberg covariant realization, while for bosons we must double it to get a reducible 4(2j+1) ‐ dimensional realization where particles and antiparticles lie in orthogonal subspaces. In short, in momentum space the FW transformation is the matrix representing a Lorentz boost times the factor (m/p _{0})^{1/2}, while in configuration space the FW transformation does not belong to the Poincaré group. The last part of the paper is devoted to getting quantum‐mechanical representations of the Galileo group as a contraction of Poincaré group representations by using mathematical methods earlier developed by Mickelsson and Niederle. The relevance of our generalized FW transformation for getting a smooth, well defined, nonrelativistic limit is a remarkable result.

Internal Galilei group for a two‐particle system
View Description Hide DescriptionTwo variable expansions of Galileian scattering amplitudes are proposed for a two‐particle system with arbitrary spins. The barycentric decomposition of such a system permits, on the one hand, to realize the associated Hilbert space in the form H=L ^{2}(R^{3}) ⊗ H_{INT}, where H_{INT} is the Hilbert space of the states of the system expressed in their CM frame, and, on the other hand, to introduce what we call the internal Galilei group G (3)_{INT} as the maximal subgroup in G (3) ×G (3) which respects the barycentric decomposition. For this group we have a natural unitary representation acting on H_{INT} induced by the PIUR of G (3) ×G (3) acting on H. The reduction of this representation with respect to the chain of subgroups G (3)_{INT} ⊆E (3)_{INT} ⊆ SU(2) ⊆ SU(1) provides the usual one variable expansions while the reduction with respect to the chain G (3)_{INT} ⊆G (3)_{0INT} ⊆ SU(2) ⊆ SU(1) provides an example of a two variable expansion.

Positivity conditions on correlation functions that imply Debye screening
View Description Hide DescriptionIn the classical statistical mechanics setting, a set of positivity conditions on certain two‐point correlation functions is exhibited that implies Debye screening for a large class of Coulomb‐like models. For example, for the model treated by Brydges, for which he has rigorously proved shielding, in a range of parameters where 〈φ^{ s }(x) J (y) 〉⩾ 0 for all x and y and all s odd, there is screening. (Alternative conditions require positivity for only two correlation functions.) Strong estimates are obtained for the rate of exponential falloff.

Generalized susceptibility of a solitary wave
View Description Hide DescriptionWe define a generalized susceptibility for solitary wave solutions of the nonlinear Klein–Gordon equation and obtain its expression in terms of the complete set of functions which arise in the linear stability analysis of the solitary wave. Explicit expressions are presented for the susceptibility of the sine‐Gordon soliton and the φ^{4} kink. Plots are presented for the long‐wave dynamic polarizability of the φ^{4} kink which have application to the response of ferroelectric domain walls to an oscillating external electric field.

Resolvent integration techniques for generalized transport equations
View Description Hide DescriptionA generalized class of ’’transport type’’ equations is studied, including most of the known exactly solvable models; in particular, the transport operator K is a scalar type spectral operator. A spectral resolution for K is obtained by contour integration techniques applied to bounded functions of K. Explicit formulas are developed for the solutions of full and half range problems. The theory is applied to anisotropicneutron transport, yielding results which are proved to be equivalent to those of Mika.

The Coulomb and Coulomb‐like off‐shell Jost functions
View Description Hide DescriptionThe off‐shell Jost functions are studied for a potential which is the sum of the Coulomb potential and an arbitrary local short‐range central potential. We derive their singular on‐shell behavior and their connection with the pure Coulomb off‐shell Jost functions. For the latter we derive a large variety of interesting explicit analytic expressions which are useful for various purposes.

Petrov type N vacuum metrics and homothetic motions
View Description Hide DescriptionPetrov type N vacuum spaces which admit an expanding and/or twisting principal null congruence and a homothetic motion are considered. It is shown that there are no such spaces which admit two Killing vectors, or one Killing vector of special type. If there are no Killing vectors present, the form of the homothetic Killing vector is restricted to one possibility.

Circular motion for a time‐asymmetric relativistic two‐body problem
View Description Hide DescriptionA formalism developed in a previous paper yields necessary and sufficient conditions and a solution for the circular motion case of the time‐asymmetric relativistic two‐body problem in which one particle responds to the retarded Liénard–Wiechert field of a second, while the second responds to the advanced field of the first. The necessary conditions contradict Künzle’s exceptional circular motion solution with zero angular momentum; consequently, zero angular momentum implies one‐dimensional motion. The limit in which the mass of one of the particles becomes infinite commutes with the nonrelativistic limit and reduces the solution properly to the circular motion solution of the relativistic one‐body Coulomb problem.

Time delay in N‐body scattering
View Description Hide DescriptionWe extend the theory of time delay to N‐body scattering. The known results relating time delay to the S matrix in the two‐body and three‐body problem suggest that these relationships are universal. Within the context of two‐Hilbert space N‐body scattering theory an abstract definition of time delay is provided. For all scattering processes initiated by the collision of two clusters a simple proof is constructed establishing the connection of time delay to the on‐shell S matrix and its energy derivatives. The definition of time delay and method of proof given here are compared with earlier approaches used in the three‐body problem.