Volume 23, Issue 12, December 1982
Index of content:

Generating functions for IR multiplicities
View Description Hide DescriptionThe construction of generating functions for multiplicities of irreducible representations from generating functions for compound characters is examined. Weyl reflection symmetry is used to simplify the procedure. Two examples involving the enumeration of SU(3) irreducible representations are discussed.

Implementation of automorphism groups in certain representations of the canonical commutation relations
View Description Hide DescriptionNecessary and sufficient conditions are given for the unitary implementability of one‐parameter unitary groups of one‐particle automorphisms of the CCR algebra in representations symplectically related to the Fock representation. The criteria become particularly simple when the one‐particle generator of the unitary group is positive and bounded away from zero; in this case the automorphism group is unitarily implementable only in the representations unitarily equivalent to the Fock representation. If the spectrum of the generator includes zero, however, the situation is more complicated; there then exist representations inequivalent to the Fock representation which admit unitary implementation of the automorphism group. It is also shown that whenever implementation of the automorphism group is possible, the implementing operators can be chosen to be a strongly continuous unitary group, guaranteeing the existence of a ‘‘second‐quantized’’ self‐adjoint generator.

Decomposition of the finite‐dimensional fermion algebra into irreducible spaces
View Description Hide DescriptionThe 2^{2n } ‐dimensional operator algebra constructed on n single‐fermion states is decomposed into irreducible tensor operator spaces with respect to three Lie subalgebras of physical interest: (i) the Lie subalgebra associated with the group SU(n) used in Hartree–Fock theory, (ii) the Lie subalgebra associated with the group SO(2n) used in Hartree–Bogoliubov theory, and (iii) the Lie subalgebra associated with the group SO(2n+1) introduced by Wybourne in atomic applications.

The lifting of an İnönü–Wigner contraction at the level of universal coverings
View Description Hide DescriptionIt is shown that, when the Borel cohomology of a connected Lie groupG is such that all projective representations can be lifted to unitary representations of the universal covering group, then any contraction of G corresponds to a contraction of its universal covering. Three theorems are stated and proved. The results apply also to the İnönü–Wigner contraction of the Poincaré group into the Galilei group.

Irreducible representations of the superalgebras type II
View Description Hide DescriptionThe representation of the orthosymplectic algebras and the other members of their class are built explicitly, with simple techniques.

Reduction of tensor products with definite permutation symmetry: Embeddings of irreducible representations of Lie groups into fundamental representations of SU(M) and branchings
View Description Hide DescriptionWe consider tensor products made out of a number of identical copies of the defining representations of Lie groups that are asymptotically free and complex. Decomposition of the tensor products into the terms with definite permutation symmetry is made by using the index sum rules and the congruence class. The results can also be used to find the branchings of SU(M) into a Lie groupG, where M is equal to the dimension of the defining representation of G. Application of our results to preon dynamics is indicated in two examples.

On a generalized Hilbert problem
View Description Hide DescriptionThe problem analyzed is to find functions f _{±}, meromorphic in C^{±}, respectively, with values that are linear operators on a Banach space, and such that their boundary values on R satisfy the equationf _{−}=ω f _{+}, where the operator‐valued function ω as well as the positions of the poles of f _{±} and the ranges of their residues are given. Uniqueness results are obtained, under certain conditions an index is proved to exist, and the determination of f _{±} is reduced to the solution of a generalization of Marchenko’s fundamental equation. The results are applied to inverse scattering and inverse spectral problems.

A class of discontinuous integrals involving Bessel functions
View Description Hide DescriptionA general theorem, which appears to be newly discovered although it is of a very classical sort, gives simple evaluations for a large class of infinite integrals containing Bessel functions in product with other suitably constrained analytic functions.

The commutant of a multiplication operator
View Description Hide DescriptionWe determine the class of all operators commuting with a multiplication operator defined by a general piecewise continuous strictly monotonic function.

Carleman embedding and Lyapunov exponents
View Description Hide DescriptionWe investigate the solutions of those autonomous systems with quadratic nonlinearities in a N‐dimensional vector space together with the solutions of their first variational equation systems by means of the Carleman embedding. An iterative procedure based on this result is developed to evaluate the Lyapunov exponents of the considered systems. We test the method by giving some results for the Lyapunov exponents of the Lorenz model.

Fractional approximations for linear first‐order differential equations with polynomial coefficients—application to E _{1}(x)
View Description Hide DescriptionA method is described to obtain fractional approximations for linear first‐order differential equations with polynomial coefficients. This approximation can give good accuracy in a large region of the complex variable plane that may include all of the real axis. The parameters of the approximation are solutions of algebraic equations obtained through the coefficients of the higher and lower powers of the variable after the substitution of the fractional approximation in the differential equation. The method is more general than the asymptotical Padé method, and it is not required to determine the power series or asymptotical expansion. A simple approximation for the exponential integral is found, which gives three exact digits for most of the real values of the variable. Approximations of higher accuracy than those of other authors are also obtained.

On the polynomial first integrals of certain second‐order differential equations
View Description Hide DescriptionIt is shown that a n y first integral of type P _{2}(ẋ)—a polynomial of degree 2 in ẋ—of the differential equationẍ=V _{ x } can be obtained from a pointlike gauge symmetry of the action A _{ L } associated to L= 1/2 ẋ^{2}+V(t,x). The same result holds for any first integral of kind P _{ n }(ẋ) when dynamical symmetries of A _{ L }polynomials in ẋ are allowed. The neccessary and sufficient conditions that V(t,x) must satisfy in order that ẍ=V _{ x } possesses a first integral of type P _{ n }(ẋ) have been obtained. These conditions reduce (when n=2) to a condition obtained by Leach. The computational advantages and difficulties which appear in order to obtain first integrals for type P _{ n }(ẋ) are also briefly discussed.

The inverse scattering problem for L C R G transmission lines
View Description Hide DescriptionThe inverse scattering problem for one‐dimensional nonuniform transmission lines with inductanceL(z), capacitanceC(z), series resistance R(z) and shunt conductance G(z) per unit length (z∈R) is considered. It is reduced to the inverse scattering problem for the Zakharov–Shabat system. It is found that one can construct from the data the following functions of the travel time x: q̃^{±}(x)=[(1/4)(d/d x)(ln(L/C))±(1/2)(R/L−G/C)] ×exp(∓∫^{ x } _{∞}(R/L+G/C)d y).

Two‐dimensional scattered fields: A description in terms of the zeros of entire functions
View Description Hide DescriptionA general description of n‐dimensional Fourier transforms is given in terms of their complex zero surfaces. The properties of these surfaces are analyzed and then applied to two‐dimensional scatteredelectromagnetic fields in the Fraunhofer region. It is shown that the properties of two‐dimensional fields differ inherently from those of one‐dimensional fields and that they lead to a reduced ambiguity for object reconstruction from intensity data. A way of estimating this ambiguity is given.

Global solution to a nonlinear integral evolution problem in particle transport theory
View Description Hide DescriptionExistence and uniqueness of the solution to a nonlinear integral evolution equation, arising in particle transport theory, is discussed and proved for any time interval [0,T]. This is pursued by a suitable application of the contracting mapping principle to the study of the nth power A ^{ n } of the relevant nonlinear inhomogeneous integral operator A.

Nonclassical fields with singularities on a moving surface
View Description Hide DescriptionFields with singularities on a moving surfaceS with boundary ∂S can be represented as distributions which have their support concentrated on S and ∂S. This paper considers such fields of the form F={ f }+λδ_{ S̃}, where { f } is the distribution determined by a field f and λδ_{ S̃} is a Dirac delta distribution with density λ concentrated on the tube S̃ swept out by the moving surface. A straightforward calculation of the distributional gradient, curl, divergence, and time derivative of such fields yields fields of the following general form: G={ g } +αδ_{ S̃} +βδ_{∂S̃} +γ∇_{ n }(⋅)δ_{ S̃}. The density α is shown to contain all the information which is customarily presented in the jump conditions for fields with singularities at a moving interface. Examples from electromagnetic field theory are presented to show the significance of the other terms { g }, βδ_{∂S̃}, and γ∇_{ n }(⋅)δ_{ S̃}.

The inverse problem of the calculus of variations applied to continuum physics
View Description Hide DescriptionNecessary and sufficient conditions for a differential system of equations to admit a variational formulation are established by having recourse to Vainberg’s theorem which provides also a systematic method for producing the sought functional. An application of the method to the Lagrangian description of fluid dynamics leads to a new variational principle which, while being fully general, reveals a hierarchy between variational approaches to fluid dynamics. Next, the method is applied in an attempt to obtain new variational formulations in various areas of research pertaining to continuum physics: waterwave models, elasticity, heat conduction in solids, dynamics of anharmonic crystals, and electromagnetism. Owing to the power of the method, relevant variational formulations are found whenever the given system allows them. The paper places particular emphasis on equations which have, or are supposed to have, soliton solutions.

Maximum of the spin‐flip cross section from unitarity and four constraints
View Description Hide DescriptionThe upper bound on the spin‐flip cross section is improved by adding a fourth constraint in a variational calculus. The total cross section, elastic cross section, the forward slope, and the backward slope of the imaginary part of the amplitude form the equality constraints. In addition the unitarity of the partial waves gives inequality constraints.

Renormalized Lie perturbation theory
View Description Hide DescriptionA Lie operator method for constructing action‐angle transformations continuously connected to the identity is developed for area preserving mappings. By a simple change of variable from action to angular frequency, a perturbation expansion is obtained in which the small denominators have been renormalized. The method is shown to lead to the same series as the Lagrangianperturbation method of Greene and Percival, which converges on KAM surfaces. The method is not superconvergent but yields simple recursion relations which allow automatic algebraic manipulation techniques to be used to develop the series to high order. It is argued that the operator method can be justified by a n a l y t i c a l l y c o n t i n u i n g from the complex angular frequency plane onto the real line. The resulting picture is one where preserved primary KAM surfaces are continuously connected to one another.

The complexification of a nonrotating sphere: An extension of the Newman–Janis algorithm
View Description Hide DescriptionA procedure given by Newman and Janis, to obtain the exterior Kerr metric from the exterior Schwarzschild metric by performing a complex coordinate transformation, is applied to an interior spherically symmetric metric. The resulting metric can be matched to the exterior Kerr metric on the boundary of the source which is chosen to be an oblate spheroid. A specific example of an interior solution for which the energy density is positive is given in detail.