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Dec 1982

Volume 23, Issue 12, pp. 2217-2595

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Generating functions for IR multiplicities

C. Bodine and R. W. Gaskell

J. Math. Phys. 23, 2217 (1982); http://dx.doi.org/10.1063/1.525310 (4 pages) | Cited 4 times

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The 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.
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02.20.-a Group theory
02.30.-f Function theory, analysis

Implementation of automorphism groups in certain representations of the canonical commutation relations

Henry A. Warchall

J. Math. Phys. 23, 2221 (1982); http://dx.doi.org/10.1063/1.525311 (8 pages) | Cited 3 times

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Necessary 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.
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02.20.-a Group theory
03.65.Fd Algebraic methods

Decomposition of the finite‐dimensional fermion algebra into irreducible spaces

A. K. Bose and A. Navon

J. Math. Phys. 23, 2229 (1982); http://dx.doi.org/10.1063/1.525312 (5 pages)

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The 22n ‐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.
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02.20.-a Group theory
05.30.Fk Fermion systems and electron gas

The lifting of an İnönü–Wigner contraction at the level of universal coverings

Ariel Fernández and Oktay Sinanoğlu

J. Math. Phys. 23, 2234 (1982); http://dx.doi.org/10.1063/1.525313 (2 pages)

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It is shown that, when the Borel cohomology of a connected Lie group G 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.
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02.20.Qs General properties, structure, and representation of Lie groups

Irreducible representations of the superalgebras type II

J.‐P. Hurni and B. Morel

J. Math. Phys. 23, 2236 (1982); http://dx.doi.org/10.1063/1.525314 (8 pages) | Cited 22 times

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The representation of the orthosymplectic algebras and the other members of their class are built explicitly, with simple techniques.
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02.20.Qs General properties, structure, and representation of Lie groups
11.30.Pb Supersymmetry

Reduction of tensor products with definite permutation symmetry: Embeddings of irreducible representations of Lie groups into fundamental representations of SU(M) and branchings

A. N. Schellekens, In‐Gyu Koh, and Kyungsik Kang

J. Math. Phys. 23, 2244 (1982); http://dx.doi.org/10.1063/1.525315 (13 pages) | Cited 4 times

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We 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 group G, 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.
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02.20.Qs General properties, structure, and representation of Lie groups
11.30.Pb Supersymmetry
11.15.-q Gauge field theories
11.55.Hx Sum rules

On a generalized Hilbert problem

Roger G. Newton

J. Math. Phys. 23, 2257 (1982); http://dx.doi.org/10.1063/1.525316 (9 pages) | Cited 19 times

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The 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 equation f−=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.
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02.30.-f Function theory, analysis

A class of discontinuous integrals involving Bessel functions

Charles Schwartz

J. Math. Phys. 23, 2266 (1982); http://dx.doi.org/10.1063/1.525317 (2 pages) | Cited 2 times

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A 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.
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02.30.-f Function theory, analysis

The commutant of a multiplication operator

A. H. Nasr

J. Math. Phys. 23, 2268 (1982); http://dx.doi.org/10.1063/1.525318 (3 pages) | Cited 1 time

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We determine the class of all operators commuting with a multiplication operator defined by a general piecewise continuous strictly monotonic function.
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02.30.-f Function theory, analysis
03.65.-w Quantum mechanics

Carleman embedding and Lyapunov exponents

R. F. S. Andrade

J. Math. Phys. 23, 2271 (1982); http://dx.doi.org/10.1063/1.525319 (5 pages) | Cited 4 times

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We 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.
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02.30.Hq Ordinary differential equations

Fractional approximations for linear first‐order differential equations with polynomial coefficients—application to E1(x)

Pablo Martin and Jorge Zamudio‐Cristi

J. Math. Phys. 23, 2276 (1982); http://dx.doi.org/10.1063/1.525320 (5 pages) | Cited 1 time

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A 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.
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02.30.Hq Ordinary differential equations
02.60.Gf Algorithms for functional approximation

On the polynomial first integrals of certain second‐order differential equations

F. G. Gascón, F. B. Ramos, and E. Aguirre‐Daban

J. Math. Phys. 23, 2281 (1982); http://dx.doi.org/10.1063/1.525306 (5 pages) | Cited 8 times

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It is shown that any first integral of type P2(math)—a polynomial of degree 2 in math—of the differential equation math=Vx can be obtained from a pointlike gauge symmetry of the action AL associated to L= 1/2 math2+V(t,x). The same result holds for any first integral of kind Pn(math) when dynamical symmetries of AL polynomials in math are allowed. The neccessary and sufficient conditions that V(t,x) must satisfy in order that math=Vx possesses a first integral of type Pn(math) 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 Pn(math) are also briefly discussed.
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02.30.Jr Partial differential equations

The inverse scattering problem for LCRG transmission lines

M. Jaulent

J. Math. Phys. 23, 2286 (1982); http://dx.doi.org/10.1063/1.525307 (5 pages) | Cited 18 times

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The inverse scattering problem for one‐dimensional nonuniform transmission lines with inductance L(z), capacitance C(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: math±(x)=[(1/4)(d/dx)(ln(L/C))±(1/2)(R/L−G/C)] ×exp(∓∫x∞(R/L+G/C)dy).
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02.30.Jr Partial differential equations
02.30.-f Function theory, analysis
84.40.Az Waveguides, transmission lines, striplines

Two‐dimensional scattered fields: A description in terms of the zeros of entire functions

I. Manolitsakis

J. Math. Phys. 23, 2291 (1982); http://dx.doi.org/10.1063/1.525308 (8 pages) | Cited 16 times

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A 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 scattered electromagnetic 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.
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02.30.Nw Fourier analysis
02.30.Fn Several complex variables and analytic spaces
42.30.Kq Fourier optics

Global solution to a nonlinear integral evolution problem in particle transport theory

V. C. Boffi and G. Spiga

J. Math. Phys. 23, 2299 (1982); http://dx.doi.org/10.1063/1.525309 (5 pages) | Cited 11 times

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Existence 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 An of the relevant nonlinear inhomogeneous integral operator A.
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02.30.Rz Integral equations

Nonclassical fields with singularities on a moving surface

David E. Betounes

J. Math. Phys. 23, 2304 (1982); http://dx.doi.org/10.1063/1.525321 (8 pages) | Cited 3 times

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Fields with singularities on a moving surface S 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 }+λδmath, where { f } is the distribution determined by a field f and λδmath is a Dirac delta distribution with density λ concentrated on the tube math 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 } +αδmath +βδ∂math +γ∇n(⋅)δmath. 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∂math, and γ∇n(⋅)δmath.
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02.30.Sa Functional analysis
03.50.De Classical electromagnetism, Maxwell equations
02.30.Jr Partial differential equations

The inverse problem of the calculus of variations applied to continuum physics

F. Bampi and A. Morro

J. Math. Phys. 23, 2312 (1982); http://dx.doi.org/10.1063/1.525322 (10 pages) | Cited 12 times

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Necessary 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: water wave 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.
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02.30.Xx Calculus of variations
02.30.Yy Control theory
47.10.-g General theory in fluid dynamics
03.50.De Classical electromagnetism, Maxwell equations
46.25.Cc Theoretical studies

Maximum of the spin‐flip cross section from unitarity and four constraints

I. A. Sakmar

J. Math. Phys. 23, 2322 (1982); http://dx.doi.org/10.1063/1.525323 (6 pages)

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The 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.
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02.30.Xx Calculus of variations
02.30.Yy Control theory
11.80.Et Partial-wave analysis

Renormalized Lie perturbation theory

E. Rosengaus and R. L. Dewar

J. Math. Phys. 23, 2328 (1982); http://dx.doi.org/10.1063/1.525324 (11 pages) | Cited 2 times

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A 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 Lagrangian perturbation 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 analytically continuing 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.
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02.40.-k Geometry, differential geometry, and topology
02.30.-f Function theory, analysis
02.50.Ey Stochastic processes

The complexification of a nonrotating sphere: An extension of the Newman–Janis algorithm

L. Herrera and J. Jiménez

J. Math. Phys. 23, 2339 (1982); http://dx.doi.org/10.1063/1.525325 (7 pages) | Cited 10 times

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A 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.
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02.40.Ky Riemannian geometries
04.20.-q Classical general relativity

Geodesic deviation and first integrals of motion

Giacomo Caviglia, Franco Salmistraro, and Clara Zordan

J. Math. Phys. 23, 2346 (1982); http://dx.doi.org/10.1063/1.525326 (7 pages) | Cited 4 times

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Generalized Killing tensors (GKT) and generalized conformal Killing tensors (GCKT) are defined as the class of totally symmetric tensor fields that generate solutions of a suitable inhomogeneous equation of geodesic deviation along arbitrary and null geodesics, respectively. A geometric interpretation of these fields as generators of Jacobi fields along arbitrary and null geodesics is also given. It is shown that well known fields such as Killing tensors, conformal Killing tensors, and geodesic collineations belong to the above classes. Finally, first integrals of geodesic motion concomitant with the existence of GKT’s and GCKT’s are determined.
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02.40.Ky Riemannian geometries
04.20.-q Classical general relativity

Cascaded Poisson processes

Kuniaki Matsuo, Bahaa E. A. Saleh, and Malvin Carl Teich

J. Math. Phys. 23, 2353 (1982); http://dx.doi.org/10.1063/1.525327 (12 pages) | Cited 9 times

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We investigate the counting statistics for stationary and nonstationary cascaded Poisson processes. A simple equation is obtained for the variance‐to‐mean ratio in the limit of long counting times. Explicit expressions for the forward‐recurrence and inter‐event‐time probability density functions are also obtained. The results are expected to be of use in a number of areas of physics.
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02.50.-r Probability theory, stochastic processes, and statistics
29.40.-n Radiation detectors
87.10.-e General theory and mathematical aspects
96.50.sd Extensive air showers

A generating integral for matrix elements of the Coulomb Green’s function

Robert Nyden Hill and Barton D. Huxtable

J. Math. Phys. 23, 2365 (1982); http://dx.doi.org/10.1063/1.525328 (6 pages) | Cited 6 times

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The generating integral Il(λ,λ′;E) =∫d3r d3r′ exp(−λr−λ′r′)(rr′)l−1 ×mathl,m(θ,ϕ)G(r,r′;E) ×Yl,m(θ′,ϕ′) is evaluated, where the Coulomb Green’s function G is the inverse (E−H)−1 with H the hydrogenic Hamiltonian −(1/2) ∇2−Zr−1 and Yl,m is a spherical harmonic. The result can be used for the evaluation of matrix elements of G with respect to wave functions of the form Yl,m(θ,ϕ)f(α)l,p(r) where f(α)l,p(r) =N(α)l,p ×exp(−ar/2)(ar)lL(α)p (ar) with L(α)p a Laguerre polynomial and N(α)l,p a normalizing factor. For general E the result is given in terms of a special case of the hypergeometric function which satisfies an inhomogeneous linear first‐order ordinary differential equation. For E=En where En=−Z2/(2n2) is a hydrogenic bound state energy, G is replaced by the generalized Green’s function (generalized inverse) and the results are given in closed form in terms of elementary functions.
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02.70.-c Computational techniques; simulations
32.90.+a Other topics in atomic properties and interactions of atoms with photons (restricted to new topics in section 32)

A direct approach to finding exact invariants for one‐dimensional time‐dependent classical Hamiltonians

H. Ralph Lewis and P. G. L. Leach

J. Math. Phys. 23, 2371 (1982); http://dx.doi.org/10.1063/1.525329 (4 pages) | Cited 82 times

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For a classical Hamiltonian H=(1/2) p2+V(q,t) with an arbitrary time‐dependent potential V(q,t), exact invariants that can be expressed as series in positive powers of  p, I(q,p,t)=∑∞n=0pnfn(q,t), are examined. The method is based on direct use of the equation dI/dt=∂I/∂t +[I,H] =0. A recursion relation for the coefficients fn(q,t) is obtained. All potentials that admit an invariant quadratic in p are found and, for those potentials, all invariants quadratic in p are determined. The feasibility of extending the analysis to find invariants that are polynomials in p of higher degree than quadratic is discussed. The systems for which invariants quadratic in p have been found are transformed to autonomous systems by a canonical transformation.
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45.05.+x General theory of classical mechanics of discrete systems

Linear Hamiltonian systems are integrable with quadratics

Huseyin Kocak

J. Math. Phys. 23, 2375 (1982); http://dx.doi.org/10.1063/1.525330 (6 pages) | Cited 2 times

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A new proof of a theorem of Williamson on the complete integrability of time‐independent, real, linear Hamiltonian differential equations with quadratic integrals is given. The sets where these integrals are functionally dependent are explicitly found.
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45.05.+x General theory of classical mechanics of discrete systems
02.20.Sv Lie algebras of Lie groups
02.30.Hq Ordinary differential equations
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