Volume 21, Issue 2, February 1980
Index of content:

A class of dissipative evolutions with applications in thermodynamics of fermion systems
View Description Hide DescriptionWe construct explicitly a special class of semigroups of completely positive maps on the CAR algebra and give an application on a model of thermal contact.

Generalized back coupling rules for the Racah algebra of Gln
View Description Hide DescriptionTwo subgroup relations for double coset matrix elements (DCME) of the symmetric group S _{ N } are derived by considering the processes of subduction and induction. The duality of outer product coupling in S _{ N } and inner product coupling in Gln identifies these as generalized back coupling rules for the Racah algebra of Gln. An iterative procedure for evaluating the DCME, once a consistent phase convention is established, is given. As an example the Racah sum formula for the Clebsch–Gordan coefficients of SU2 is derived from consideration of S _{ N } coupling only.

Finite subgroups of the generalized Lorentz groups O(p,q)
View Description Hide DescriptionAn algorithm is developed which permits us to construct all the finite subgroups of the generalized Lorentz groups O(p,q) up to an O(p,q) conjugation. The application of this algorithm to the Lorentz group O(3,1) is outlined, and the full list of finite subgroups compiled.

All solutions to a nonlinear system of complex potential equations
View Description Hide DescriptionThis work employs the powerful geometric methods previously developed in order to determine all solutions of the nonlinear system (∇γ)^{2}≡∇γ⋅∇γ=f (γ), (∇^{2}γ)^{2}−(1/2)(N+2) f′ (γ) ∇^{2}γ+(1/4)(N+1)[f′ (γ)]^{2} −(1/2) N f (γ) f\ (γ) =−N∇γ⋅∇ (∇^{2}γ), where f (γ) is an arbitrarily assigned function and N is an arbitrary constant. The circumstances are determined under which compatible solutions exist, not only when γ is real, but also when γ is complex, and all of the corresponding solutions are found. This is done by referring the system of equations to a set of coordinates based on the (real or complex) equipotential surfaces of constant γ. The relationship between the solutions and the geometry of the equipotential surfaces is examined, and a close association is discovered between the set of allowable equipotential surfaces and the class of surfaces of constant total (Gaussian) curvature. These results are analogous to those in Collins, Math. Proc. Camb. Phil. Soc. 80, 165–87 (1976), where the system under study was associated with equipotentials of constant radii of principal normal curvature. The geometrical method used throughout is remarkable in that it yields a knowledge of a l l solutions of the given system. It thereby offers the possibility of application to a wide variety of fields in physics where similar systems of equations are encountered.

Complex potential equations, special relativity, and complexified Minkowski space–time
View Description Hide DescriptionWe obtain a number of results on null geodesic congruences in Minkowski space–time M. It is first shown that the only time‐invariant hypersurface‐orthogonal shear‐free null geodesic congruences in M are those that generate either null hyperplanes or null cones. This result is derived in two interesting and quite distinct ways. One method of proof uses Kerr’s theorem. The other method proceeds by showing that the equations specifying the class of congruences can be written in the form (∇γ)^{2}≡∇γ⋅∇γ=1, and (∇^{2}γ)^{2}=−2∇γ⋅∇ (∇^{2}γ), where γ is a potential and the operators ∇ and ∇^{2} refer to 3‐dimensional Euclidean space. The complexified version of this system of equations has been studied previously by the author [preceding paper, J. Math. Phys. 21, 240 (1980)]. Hypersurface‐orthogonal shear‐free null geodesic congruences in complexified Minkowski space–time are then investigated, and a close association is discovered between the set of such congruences and the set of (generally, non‐hypersurface‐orthogonal) shear‐free null geodesic congruences in M.

Bäcklund transformations and the symmetries of the Yang equations
View Description Hide DescriptionUsing the terminology of Jet bundles, we determine generalized symmetries of the Yin equations f (f _{ y ȳ}+f _{ z z̄})−f _{ y } f _{ ȳ} −f _{ z } f _{ z̄}−e _{ y } g _{ ȳ}−ε_{ z } g _{ z̄ }=0, f (e _{ y ȳ}+e _{ z z̄})−2e _{ y } f _{ ȳ} −2e _{ z } f _{ z̄}=0, f (g _{ y ȳ}+g _{ z z̄})−2g _{ ȳ}−2g _{ ȳ} f _{ y } −2g _{ z̄} f _{ z }=0, which are equivalent to the self‐dual Yang–Mills equations in a particular gauge. The Bäcklund transformations of Corrigan e t a l. are derived and discussed as generalized symmetries.

An improvement of Watson’s theorem on Borel summability
View Description Hide DescriptionWatson’s theorem, which gives sufficient conditions for Borel summability, is not optimal. Watson assumes analyticity and uniform asymptotic expansion in a sector ‖argz‖<π/2+ε, ‖z‖<R, with ε≳0; in fact, only the circular region Re(1/z) ≳1/R is required. In particular, one can take ε=0. This improved theorem gives a necessary and sufficient characterization of a large class of Borel‐summable functions. I apply it to the perturbation expansion in the φ_{2} ^{4}quantum field theory.

The Jost solutions for general Gaussian potentials
View Description Hide DescriptionThe explicit Born series which represent the regular solutions (of the Schrödinger equation), the Jost solutions, and the Jost functions analytic in some domains of complex wave numbers and angular momenta are found for general potentials of the Gaussian type expressed by the Stiltjes integral. When finding their form we make use of the analogy with the corresponding solutions for general potentials of the Yukawa type, which is based on the special representations of the Bessel and Hankel functions (purely kinematicsolutions) in either case. Some other relations are also derived.

Partial inner product spaces. III. Compatibility relations revisited
View Description Hide DescriptionThis paper continues our systematic study of partial inner product spaces. We show here that a linear compatibility relation on a vector space V is characterized by special families of vector subspaces of V, called involutive coverings, and vice versa. This result provides the link between partial inner product spaces, defined in an intrinsic way, and various concrete structures, such as rigged or nested Hilbert spaces. Given a linear compatibility, generating sets (’’rich subsets’’) are discussed, and several examples are worked out. Finally, we introduce an order relation among all linear compatibilities on the same vector space.

A modified asymptotic Padé method. Application to multipole approximation for the plasma dispersion function Z
View Description Hide DescriptionAn improved asymptotic Padé method is presented. The approximations expressed as rational functions are developed both in a power series and asymptotic expansions. Identifying these developments with those for the exact function, the approximated rational functions are obtained. Three‐ and four‐pole approximations for the plasma dispersion function have been determined with this method. Our approximations (Z _{ lm}) give closer results to the exact function than all the published ones.

Dynamical symmetry breaking in the Lewis–Riesenfeld oscillator
View Description Hide DescriptionDynamical realization is given to the generator responsible for breaking the usual SU(3) invariance symmetry in the case of the Lewis–Riesenfeld time‐dependent oscillator:H (t) = 1/2 J^{3} _{ i }[p _{ i } p _{ i }+w ^{2}(t) q _{ i } q _{ i }]. This is achieved by finding a simple expression for the time‐dependent Hamiltonian in the interaction picture: H=H _{o}+g H _{1}. The breaking interaction is shown to transform as (6⊕6̄) of SU(3). Time‐dependent dilations relate the broken Hamiltonian to the Lewis–Riesenfeld form. This scaling generates the algebra of the Weyl group and its role in oscillator noninvariance symmetries is considered.

The physical optics method in electromagnetic scattering
View Description Hide DescriptionThe purpose of this work is to analyze the physical optics method as applied to electromagneticscattering theory and to point out its physical and mathematical drawbacks. The main conclusions are (1) that the boundary values assumed by physical optics lead to electromagnetic fields that do not satisfy the finiteness of energy condition and, as a consequence, that integral representations of these fields cannot be obtained via the divergence theorem; (2) that the commonly accepted representations are not solutions of the physical optics problem because they fail to reproduce the assumed discontinuities of the fields on the scatterer. Despite the above conclusions, the present work should not be construed as an attempt to discredit the method but rather as an effort toward a better understanding of it. As it is well known, there have been a number of occasions in which physical optics has yielded quite satisfactory results.

The complete symmetry group of the one‐dimensional time‐dependent harmonic oscillator
View Description Hide DescriptionThe five invariants for the time‐dependent one‐dimensional harmonic oscillator Hamiltonian are constructed. Using the linear transformation to the time‐dependent oscillator Hamiltonian, the five invariants for the latter are obtained. The differential operators which generate the dynamical symmetry of this Hamiltonian have the same commutator relations as these of the time‐independent problem. An additional three operators are obtained using the method of extended Lie groups and have the same properties as those for the time‐independent problem. Thus the complete dynamical symmetry of the time‐dependent problem is the eight‐parameter Lie group SL(3,R).

On the infrared problem in nonrelativistic quantum electrodynamics
View Description Hide DescriptionFollowing a suggestion of Hepp and Lieb, it is shown rigorously that the infrared divergences which occur in the problem of thermodynamic stability of a system of atoms (with a finite number of levels) interacting with an ultraviolet‐cutoff quantized radiation field are entirely removed upon inclusion of the term in A^{2} in the Hamiltonian, in the special case where the atoms are placed at the points of a regular lattice.

Two sided estimates of the scattering amplitude for low energy
View Description Hide DescriptionEstimates and iterative processes for calculating the scattering amplitude for k=0 are given.

A study of the completeness properties of resonant states
View Description Hide DescriptionThe completeness properties of the discrete set of bound state, virtual states, and resonant states of a Hamiltonian H is investigated, where H describes a system in which a single nonrelativistic‐spinless particle moves in a central cutoff potential. A limited form of completeness is obtained. It is shown that the convergence of the resulting ’’completeness series’’ is very sensitive to the detailed mathematical structure of the potential.

Inverse scattering problems in higher dimensions: Yang–Mills fields and the supersymmetric sine‐Gordon equation
View Description Hide DescriptionIt is shown that the notion of a prolongation structure can be extended to higher dimensions and used to determine inverse scattering problems. The relationship to generalized Lax representations is also considered. The method is illustrated on the self‐dual Yang–Mills equations. A generalization to include Grassman algebra valued variables is shown to provide a scattering problem for the supersymmetric sine‐Gordon equation.

Numerical computation of off‐shell two‐body Coulomb amplitude values
View Description Hide DescriptionPractical methods of computing numerical values for off‐shell two‐body Coulomb scattering amplitudes are developed. The methods are based on well‐known analytic representations of the amplitudes. The results of a comparative computational study of the different methods are presented. It is found that the methods produce numerical values sufficiently precise for use in treating many‐body problems formulated in terms of two‐body amplitudes.

Remarks on characteristic classes of four‐dimensional Einstein manifolds
View Description Hide DescriptionUsing the decomposition of the curvature tensor by means of the dual * operator, several useful expressions for characteristic classes in some kinds of Einstein manifolds are obtained. Consequently, some topological restrictions of the underlying manifolds are considered.

Equilibrium of slowly rotating relativistic fluids
View Description Hide DescriptionThe equilibrium conditions of a relativistic fluid with nonzero viscosity and heat conduction coefficients are known to reduce to Einstein’s equations for a barotropic perfect fluid in rigid motion. We consider here the linearization of these equations on a static spherically symmetric background and show that the solution space is three‐dimensional (parametrized by the angular velocity vector, for example), provided the exterior vacuum region is asymptotically Euclidean and the equation of state ρ=ρ (p) (satisfying ρ⩾p⩾0 and d p/dρ≳0) is fixed, as well as the central values of the pressure and the gravitational potential. In the exterior region this solution agrees with the Kerr solution, linearized on the Schwarzschild background. This result is the first step towards proving a certain uniqueness of the possible equilibrium configurations of slowly rotating relativistic fluids. It is obtained using invariantly defined global conditions, without assuming the existence of particular coordinate systems.