Index of content:
Volume 22, Issue 6, June 1981

On the irreducibility of projective corepresentations—Application to double Shubnikov point groups
View Description Hide DescriptionAn irreducibility criterion for projective corepresentations is stated and specified to projective corepresentations of double Shubnikov point groups, where the choice of the considered factor systems is influenced by the representation theory of double Shubnikov space groups.

Objects for the symmetric group
View Description Hide DescriptionThe permutations, group multiplication tables, decomposition of permutations into neighboring transpositions, representation matrix elements, basis states, and basis state operators have been constructed for the symmetric groups S(n), n = 1,...,5. Part of the results have been relegated to PAPS. The algorithm for generating the permutations of the symmetric groups is discussed.

Dimension and character formulas for Lie supergroups
View Description Hide DescriptionA character formula is derived for Lie supergroups. The basic technique is that of symmetrization and antisymmetrization associated with Young tableaux generalized to supergroups. We rewrite the characters of the ordinary Lie groups U(N), O(N), and Sp(2N) in terms of traces in the fundamental representation. It is then shown that by simply replacing traces with supertraces the characters of certain representations for U(N/M) and OSP(N/2M) are obtained. Dimension formulas are derived by calculating the characters of a special diagonal supergroup element with (+1) and (−1) eigenvalues. Formulas for the eigenvalues of the quadratic Casimir operators are given. As applications, the decomposition of a representation into representations of subgroups is discussed. Examples are given for the Lie supergroup SU(6/4) which has physical applications as a dynamical supersymmetry in nuclei.

Infeld–Hull factorization, Galois–Picard–Vessiot theory for differential operators
View Description Hide DescriptionThe Infeld–Hull theory of factorization of second order differential operators is related, on the one hand, to the classical Picard–Vessiot theory and, on the other hand, to work by the author, Gel’fand, and Kirillov on Lie algebras contained in the field of fractions or algebraic extensions of the enveloping associative algebra of a Lie algebra.

Isomonodromy as a curvature‐zero condition
View Description Hide DescriptionThe ’’isomonodromy deformation’’ condition is expressed as a curvature‐zero condition, thus bringing to the foreground the relations to other approaches in the theory of nonlinear waves.

Factorization of operators.II
View Description Hide DescriptionWe extend the methods of a previous paper [J. Math. Phys. 21, 2508 (1980)], factorizing the general, scalar, third‐order differential operator, and obtain a Miura transformation for the Boussinesq equation. We give a general factorized eigenvalue problem. We also give a Hamiltonian structure associated with the factorized eigenvalue problem. We derive several isospectral flows, some of Klein–Gordon type.

The lump solutions and the Bäcklund transformation for the three‐dimensional three‐wave resonant interaction
View Description Hide DescriptionA Bäcklund transformation is found for the three‐dimensional three‐wave resonant interaction, and from it, N‐lump exact solutions may be constructed. The one‐lump solution is analyzed in detail, and it is shown that it describes such effects as pulse decay, upconversion, and explosive instabilities, all in three dimensions.

Power series expansion for a crossing‐symmetric vertex function
View Description Hide DescriptionStarting from a nonlinear system of coupled integral equations for the vertex parts which are irreducible with respect to the s‐, t‐, and u‐channel, a power series is found for the full vertex function in terms of the vertex part Γ^{0} which is irreducible with respect to all channels. The problem of finding an expression in closed form remains unresolved, since higher‐order products in Γ^{0} are nonassociative. In a simplified model, the mathematical properties of the series seem to allow successive rational approximations.

Construction of orthonormal vector sets for atoms and molecules by means of recursive variation
View Description Hide DescriptionIt is shown using a recursive variation method that the Schmidt orthonormalized vectors are optimal in the sense that the least squared distances are minimized, and that the use of maximal squared overlaps yields a Schmidt canonical orthonormalized vector set which corresponds to the Löwdin canonical vector set. By introducing the Krylov sequence, the method is applied to the derivation of a variational expression for the Lanczos vectors which tridiagonalize any self‐adjoint operator. The repeated use of two‐step variations is shown to derive a pseudopotential which is useful for the calculations of correlated pair functions.

Comments on ’’Higher order modified potentials for the effective phase integral approximation’’
View Description Hide DescriptionIt is pointed out that the comparison made in the paper by Floyd, ’’Higher order modified potentials for the effective phase integral approximation,’’ between the expansions introduced by Floyd and by the present authors, respectively, is inadequate as a test of the relative merits of the two expansions. In his comparison Floyd used our u n m o d i f i e d phase‐integral approximations, but for the case considered one should instead use our consistently m o d i f i e d phase‐integral approximations. The numerical results then obtained are in the present note compared with the numerical results obtained from the expansion derived by Floyd.

Two‐sided Padé approximations for the plasma dispersion function
View Description Hide DescriptionTwo‐sided Padé approximations are determined for the plasma dispersion function. An exact closed form solution is given. The convergence problem is considered with regard to the real axis. Some applications from the theory of charged particle beams are also given

The elastic pendulum: A nonlinear paradigm
View Description Hide DescriptionA pendulum with an elastic instead of an inextensible suspension is the simplest realization of an autonomous, conservative, oscillatory system of several degrees of freedom with nonlinear coupling; it can also have an internal 1:2 resonance. A fairly complete study of this system at and near resonance is here undertaken by means of the ’’slow‐fluctuation’’ approximation which consists in developing the x ^{2} y‐type interaction into a trigonometric polynomial and keeping only the term with the slowest frequency. Extensive computations showed that up to moderately large amplitudes the approximate solutions were virtually as accurate as numerical integrations of the exact equations of motion. The slow‐fluctuation equations of motion can be completely integrated by quadratures. Explicit solutions for amplitudes and phases are given in terms of elliptic functions, and can be linked to initial conditions. There exist two branches of purely periodic, harmonic, constant‐amplitude motions which are orbitally stable but Liapunov unstable. The pure suspension motion is Liapunov unstable and remains orbitally stable only up to and including a critical amplitude; the standard ’’method of variational equations’’ leads to a slightly different stability criterion but is shown to be unreliable. In the dynamical neighborhood of the unstable pure suspension mode are motions which convert to it after infinite time. When a motion has an amplitude modulation minimum at or near zero, a phase reversal of the suspension takes place which is shown to be an artefact inherent in the description in terms of amplitudes and phases. In addition there is in the pendulum (but not in the exactly soluble system having the slow–fluctuation Hamiltonian) a fast phase transient which vitiates the slow‐fluctuation technique for a few periods around the suspension amplitude minimum; this is the only restriction on the method. An appendix outlines formal isomorphisms between the elastic pendulum and the process of second‐harmonic generation in nonlinear optics.

On the conservation laws in the theory of fields in Finsler spaces
View Description Hide DescriptionSome conservation laws are obtained for the fields in some special Finsler spaces such as scalar curvature space, locally Minkowski space, etc.

On the theory of fields in Finsler spaces
View Description Hide DescriptionFinsler space is a metric space in which the line‐element (x, y), instead of the point (x), is chosen as the independent variable, where the vector y is regarded as the internal variable associated with each point x. Therefore, the y‐dependence characterizes essentially the theory of fields in Finsler spaces and the Finsler field shows many peculiar features depending on y. So, in this paper, with the aid of the theory of special Finsler spaces such as tangent Riemannian space, locally Minkowski space, etc., some physical aspects carried by y are considered by taking into account the intrinsic behavior of y.

Schrödinger equation with time‐dependent boundary conditions
View Description Hide DescriptionA Schrödinger equation for a well potential with varying width is studied. Generalized canonical transformations are shown to transform the problem into a time‐dependent harmonic oscillator problem submitted to fixed boundary conditions. This transformed problem is solved by a perturbation technique and gives the evolution of the average energy of the system according to the motion of the well. Motions corresponding to a renormalization or compaction group are shown to be solvable by separation of variables.

Exact propagators for multidimensional quadratic Hamiltonians—Initial‐value approach
View Description Hide DescriptionA practical method suitable for both analytic and numerical computation of propagators for multidimensional quadratic Hamiltonians in presented. The traditional two‐end‐point problem is replaced by a manageable initial‐value problem. As a by‐product, the classical path and action are also obtained as a solution to an initial‐value problem. The method is illustrated analytically as well as numerically by solving for the motion of a particle in a constant magnetic field.

High‐energy behavior of phase shifts for scattering from singular potentials
View Description Hide DescriptionThe high‐energy behavior of the Jost function in nonrelativistic potential scattering theory is studied for potentials, strongly singular and repulsive at the origin. To be able to give the result in an explicit form, we specialize to the S wave and to pure inverse power potentials. However, the method works for a large class of potentials, and we report briefly on some other cases. We use an asymptotic method, which can be described as a generalization of the JWKB method to arbitrary order, with rigorous error bounds. Some numerical results are given.

Multipole expansion of stationary asymptotically flat vacuum metrics in general relativity
View Description Hide DescriptionA multipole expansion scheme is introduced for a wide class of stationary, asymptotically flat, vacuum solutions of Einstein’s equations using the conformal techniques of Geroch and Hansen. An intrinsic choice of the conformal factor and suitable asymptotic flatness conditions enable one to express the rescaled gravitational mass and angular momentum potentials and the rescaled spatial metric as power series in normal coordinates around a point Λ representing the spatial infinity on the conformal manifold. The coefficients of this expansion are certain nonlinear combinations of the Hansen multipole moments. As an example the Schwarzschild metric is discussed in the present framework.

On almost causality
View Description Hide DescriptionThe almost causal precedence concept proposed by Woodhouse and a causality axiom based on it are analyzed. Properties of the almost future are examined and the above causality axiom is shown to be deducible for a causally continuous space–time. It is shown to be coinciding with known causality conditions for a reflecting space–time. Also the equivalence of various causality conditions for a reflecting space–time and a reduction of the causal continuity axiom are obtained. It is next shown that the above causality axiom is not stable under metric perturbations. Then the Seifert future concept has been analyzed and the Woodhouse causality is proved to be strictly weaker than the stable causality condition. It is concluded that as far as the uncertainties in the form of metric perturbations are concerned, the Seifert future is useful but not the almost future.

Null Killing vectors
View Description Hide DescriptionSpace–times admitting a null Killing vector are studied, using the Newman–Penrose spin coefficient formalism. The properties of the eigenrays (principal null curves of the Killing bivector) are shown to be related to the twist of the null Killing vector. Among the electrovacs, the ones containing a null Maxwell field turn out to belong to the twist‐free class. An electrovac solution is obtained for which the null Killing vector is twisting and has geodesic and shear‐free eigenrays. This solution is parameterless and appears to be the field of a zero‐mass, spinning, and charged source.