Volume 21, Issue 6, June 1980
Index of content:

The unitary multiplicity‐free representations of SL_{4}(R)
View Description Hide DescriptionThose unitary representations of S̄L̄_{4}$̄(̄R̄)́bar)= are classified which contain each irreducible S̄Ō(̄4̄)́bar)= representation at most once.

The representations of spl(2,1)—an example of representations of basic superalgebras
View Description Hide DescriptionThe example of spl(2,1) illustrates some important properties of the representations of basic superalgebras. Typical representations are most similar to the representations of semisimple Lie algebras. We argue that nontypical representations are a degenerate type of representation. The structure of not fully reducible nontypical representations is discussed in detail. As opposed to semisimple Lie algebras, spl(2,1) also admits representations with nondiagonal Cartan subalgebra. There is a one‐to‐one correspondence between the representations of spl(2,1) and those of the generalized superalgebra osp(2,0,1,1). The osp(2,0,1,1) counterparts of nontypical irreducible representations of spl(2,1) are unfaithful.

The tensor product of two irreducible representations of the spl(2,1) superalgebra
View Description Hide DescriptionThe Clebsch–Gordan series and the unnormalized Clebsch–Gordan coefficients for the tensor product of two irreducible representations of spl(2,1) are given. Special emphasis is laid on the discussion of the structure of the series when it contains several nontypical representations. Thereby a better understanding of the ’’degenerate’’ properties of nontypical representations is gained.

Fock space representations of the Lie superalgebra A(0,n)
View Description Hide DescriptionAn infinite class of finite‐dimensional irreducible representations and one particular infinite‐dimensional representation of the special linear superalgebra of an arbitrary rank is constructed. For every representation an orthonormal basis in the corresponding representation space is found, and the matrix elements of the generators are calculated. The method we use is similar to the one applied in quantum theory to compute the Fock space representations of Bose and Fermi operators. For this purpose we first introduce a concept of creation and annihilation operators of a simple Lie superalgebra and give a definition of Fock‐space representations.

The theory of spinors via involutions and its application to the representations of the Lorentz group
View Description Hide DescriptionA simple theory of the spinor representations of the complex orthogonal group O(d,C) in the d‐dimensional Euclidean spaceV ^{(d)} is presented via a basic lemma on involutional transformations and Cartan’s theorem on O(d,C). The arbitrary gauge factors of the representations are reduced to ± signs by introducing appropriate phase conventions. The concept of an axial involution is introduced. The plane rotations in V ^{(d)} are introduced and used to construct the representations of the proper orthogonal group O^{+}(d,C). The Lorentz group is treated as a subgroup of O(4,C). The general expression for the basic 2×2 irreducible representations A(L _{0}) of the proper orthochronous Lorentz group G(L _{0}) is obtained by direct reduction of the 4×4 spinor representation S(L _{0}) by means of the basic lemma on the involutional transformations. It is completely parameterized by the angle and the axis of the spacial rotation and by the velocity of the pure Lorentz transformation. The finite dimensional irreducible representations of the Lorentz group G(L) are discussed. The transformations of electro‐magnetic field under G(L) are discussed in the most general form.

Minimal and centered graded spin‐extensions of the SL(3,R) algebra
View Description Hide DescriptionWe construct the minimal‐size infinite‐dimensional ’’supergauge’’ algebras containing sl(3,R) and spinorial generators, with the possibility of having a central term as in the Heisenberg algebra. The results describe the spin‐excitations of a membrane and may provide [when extended to sl(4,R)] the framework for an infinitely supersymmetric (renormalizable?) gravity.

A symmetry approach to exactly solvable evolution equations
View Description Hide DescriptionA method is developed for establishing the exact solvability of nonlinear evolution equations in one space dimension which are linear with constant coefficient in the highest‐order derivative. The method, based on the symmetry structure of the equations, is applied to second‐order equations and then to third‐order equations which do not contain a second‐order derivative. In those cases the most general exactly solvable nonlinear equations turn out to be the Burgers equation and a new third‐order evolution equation which contains the Korteweg‐de Vries (KdV) equation and the modified KdV equation as particular cases.

Evolution of a stable profile for a class of nonlinear diffusion equations. III. Slow diffusion on the line
View Description Hide DescriptionWe show that the nonlinear diffusion equation ∂n/∂t =∂^{2}(n ^{1+δ}/∂x ^{2} with compact initial data on −∞<x<∞ can be transformed into another nonlinear diffusion equation α(∂ϑ∂t)=ϑ^{1+α} ×∂^{2}ϑ/∂?^{2} on a fixed finite interval of the ? axis. Thus, the original moving boundary problem is transformed into a fixed boundary problem. The new form of the equation has advantages both analytically and computationally as the examples illustrate. Linear stability analysis for the transformed equation is straightforward whereas the moving boundaries of the original problem complicate the analysis for that case. The advantages of the resulting computational algorithm for solving the moving boundary problem are also discussed. A nonlinear Rayleigh–Ritz quotient and a Lyapunov functional are shown to be bounded, monotonically decreasing functions of time. Both functionals give added insight into the mathematical character of the diffusion process.

A Hilbert–Padé method for multipole approximations. Application to the Gaussian function
View Description Hide DescriptionA method is developed where a Hilbert transform is combined with an asymptotic Padé method in order to obtain good multipole approximations for functions whose power series have a large radius of convergence. This method has been used to find two‐ to eight‐pole approximations for the Gaussian function.

Existence and completeness of the Möller wave operators for radial potentials satisfying ∫_{0} ^{1} r‖v(r)‖d r+∫_{1} ^{∞}‖v(r)‖d r< ∞
View Description Hide DescriptionWe give an elementary proof of the existence of the (three‐dimensional) Möller wave operators and the unitarity of the S operator (weak asymptotic completeness) for radial Kato potentials v(r) satisfying F_{0} ^{1} r‖v(r)‖d r+F_{1} ^{∞}‖v(r)‖d r< ∞.

Jet bundles and path structures
View Description Hide DescriptionThe analysis of path structures is formulated in terms of jet bundles with particular emphasis on the transformation laws and symmetry properties of geodesic path structures. The role played by geodesic path structures in the constructive axioms of Ehlers, Pirani, and Schild for GRT is discussed and it is shown that these axioms are decidable.

A global theory of supermanifolds
View Description Hide DescriptionA mathematically rigorous definition of a global supermanifold is given. This forms an appropriate model for a global version of superspace, and a class of functions is defined which corresponds to superfields. This new construction is compared with several pre‐existing definitions of supermanifold and graded manifold; it is shown to include all these definitions and to go beyond them, particularly in admitting the possibility of nontrivial topology in the anticommuting sector. Local differential geometry and potential applications to supergravity are considered.

Extended Numerov method for the numerical solution of the Hartree–Fock equations
View Description Hide DescriptionTo match the intrinsic accuracy of an expanded basis set solution of atomic orbitals by numerical orbitals requires integration formulas of comparable intrinsic accuracy. Integration of the coupled, integro‐differential, Hartree–Fock equations is most easily done using Numerov’s formula. without increasing the number of mesh points, one must extend the basic formula to include more points. Herein is presented the derivation of the general (2N+1)‐point Numerov formula, including the error term, as well as suggestions for numerical determination with specific formulas of 3‐, 5‐, 7‐, and 9‐ point fits.

An application of Ray–Reid invariants
View Description Hide DescriptionThe formalism of Ray–Reid invariants is applied to a certain nonlinear system, first treated by Reid using a different approach. In addition, a generalization of the Ray–Reid invariant is given.

Gauge theory in Hamiltonian classical mechanics: The long‐range fields
View Description Hide DescriptionThe generalization of a previous work leads us to a gauge theory in Hamiltonian classical mechanics whose elements are: First, the infinitesimal canonical transformations considered as gauge transformations and, second, an infinite sequence of gauge potentials. A hierarchy of approximation orders allows the physical interpretation of this formalism. At the zero order we obtain the electromagnetic field, at the first order the electromagnetic and gravitational fields. At the second order a new field is added to the former ones. We have studied the new physical features involved by this hypothetical field, in the motion of a classical particle. In the approximation of a Keplerian motion, we have given the period. This last result could be eventually used to test this theory.

Coherent states and projective representation of the linear canonical transformations
View Description Hide DescriptionUsing a family of coherent state representations we obtain in a natural and coordinate‐independent way an explicit realization of a projective unitary representation of the symplectic group. Dequantization of these operators gives us the corresponding classical functions.

The hydrogen atom: Quantum mechanics on the quotient of a conformally flat manifold
View Description Hide DescriptionThe regularization of the Kepler problem proposed by Kustaanheimo and Stiefel provides an example of quantum mechanics on the quotient of a conformally flat manifold.

Decaying systems with degenerate Livsic matrix
View Description Hide DescriptionThe spectral analysis given by Wong for the resolvent of a non‐self‐adjoint operator with arbitrary multiplicity is utilized for the description of the time evolution of an unstable system. After studying the case for which the operator is independent of the resolvent variable z, the Wong analysis is extended to the physically interesting case for which the operator depends on z. The case of infinite multiplicity is treated, and it is found that the flow of probability through the generalized eigenstates is analogous to the approach to equilibrium in statistical mechanics.

Phase‐integral calculation of physically important quantities for nonrelativistic bound s states of the linear central potential
View Description Hide DescriptionEnergy levels, normalization factors, quantal expectation values, and probability densities at the origin for nonrelativistic bound s states of the linear central potential are calculated by means of phase‐integral formulas, given by N. Fröman in two recent papers. The accuracy of the phase‐integral method is exhibited by a numerical comparison with exact results. During the last few years the potential in question has been widely used as a model potential describing quark confinement in heavy mesons.

Prolongation structure for a nonlinear equation with explicit space dependence
View Description Hide DescriptionA nonlinear Schroedinger equation with a term depending explicitly on a space variable, e.g., iψ_{ t }+ψ_{ x x }+(−2αx+2‖E‖^{2}) ψ=0 with ψ=E e ^{ iφ} has been treated in the language of differential forms and prolongation. The inverse scattering equations previously invented by Liu and Chen are obtained. A unique feature of the analysis is explicit space–time dependence of the pfaffian forms.