Volume 21, Issue 7, July 1980
Index of content:

A note on the symmetries of the 3j and 6j coefficients. II
View Description Hide DescriptionThe partition of the symmetries of the 3j and 6j coefficients according to the possible expressions for the number of terms in their series representations are discussed. The canonical parametrization of the 3j and 6j coefficients introduced by Lockwood are also discussed.

Wigner–Eckart theorem for an arbitrary group or Lie algebra
View Description Hide DescriptionCircumstances sufficient for the validity of the Wigner–Eckart theorem are analyzed and the theorem is proved under very mild assumptions. The proof is purely algebraic and shows the theorem to be a direct consequence of Schur’s lemma and complete reducibility of tensor product of irreducible representations.

On Gel’fand states of representations of U(n) and the Gel’fand lattice polynomials
View Description Hide DescriptionThree theorems are proved concerning the Gel’fand lattice polynomials (GLP) introduced by Gazeaux, Dumont‐Lepage, and Ronveaux. In these theorems, subdeterminant initial conditions are assumed. (1) The Gel’fand states are linear combinations of GLP corresponding to left Gel’fand lattices. (2) The set of all GLP is a basis for the space of polynomial functions of the elements of a matrix. (3) If the left Gel’fand lattices are arranged in a natural order, then each Gel’fand state is a linear combination of the corresponding GLP and earlier ones in the same order. Finally, a formula is given for computing individual GLP.

Micu‐type invariants of exceptional simple Lie algebras
View Description Hide DescriptionA formula simplifying the quartic trace in exceptional simple Lie algebras is derived and used for expressing higher order Micu‐type invariants as polynomials in the second order Casimir invariant.

Addition of complex angular momentum operators
View Description Hide DescriptionRepresentations of the components of complex angular momentum operator have been carried out in three different subspaces of the space in which the generators of proper, orthochronous homogeneous Lorentz group operate. Addition of complex angular momentum operators have been derived in the canonical basis of their eigenvectors.

Ermakov systems, nonlinear superposition, and solutions of nonlinear equations of motion
View Description Hide DescriptionWe report several important additions to our original discussion of Ermakov systems. First, we show how to derive the Ermakov system from more general equations of motion. Second, we show that there is a general nonlinear superposition law for Ermakov systems. Also, we give explicit examples of the nonlinear superposition law. Finally, we point out that any ordinary differential equation can be included in many Ermakov systems.

On the intrinsic geometry of certain nonlinear equations: The sine‐Gordon equation
View Description Hide DescriptionThe classical relation between two‐dimensional spaces of constant curvature and certain nonlinear partial differential equations is formulated in group‐theoretic terms by means of the underlying semisimple isometry group. Rather than working with the metric and curvature in the given constant curvature space, it is then possible to consider the equivalent system consisting of a pair of first‐order partial differential equations in flat space for two so(2,1) vectors satisfying a pair of SO(2,1)‐invariant algebraic constraints. Such equations determine a SL(2,R) principal bundle with flat connection. The construction is carried out in detail for the case of the sine‐Gordon equation. The connection is shown to give rise to a spectral problem of the inverse scattering type by means of a gauge transformation. Bäcklund transformations are shown to be automorphisms of the connection characterized by a certain so(2,1) null vector. The generation of solutions of the nonlinear equation from known ones is seen to be determined by gauge transformations leaving invariant a fixed set of gauge conditions.

Bäcklund transformations in several variables
View Description Hide DescriptionA differential geometric method of constructing Bäcklund transformations for a second‐order partial differential equation in n independent variables (n arbitrary) is developed and several examples worked out.

Elementary solutions of the linear transport equation for continuously varying spatial media
View Description Hide DescriptionWe demonstrate that it is possible to construct elementary solutions (eigenfunctions) of the linear transport equation for certain types of continuously varying spatial media. In general, both discrete and continuum modes result, which appear to be complete on the half‐range. A detailed analysis is given for an ’’exponential’’ medium, including numerical results and a half‐range completeness proof. A ’’linear’’ medium is also considered. A general method is presented for constructing, jointly, the spatial variation of a medium and the corresponding functional forms of the eigenfunctions. Our results represent a partial generalization of the singular eigenfunction technique to media with continuous spatial variation.

Solutions of the sine‐Gordon equation in higher dimensions
View Description Hide DescriptionExact solutions are derived for the three‐dimensional and four‐dimensional sine‐Gordon equation [∇^{2}−∂^{2}/(c ^{2}∂t ^{2})]χ =sinχ. The principal tools in the derivation are a new Bäcklund transformation and the appropriate generating formulas which allow us to generate an infinite number of real classical solutions. Three‐dimensional computer plots depicting the time evolution are presented (a) for the two‐soliton and four‐soliton solutions of the sine‐Gordon equation, and (b) for the three‐wave interaction of the associated sine‐Gordon equation.

Solution of the Chandrasekhar H‐equation by Newton’s Method
View Description Hide DescriptionFor any value of the parameter c, c≠1, for which a solution to the Chandrasekhar H‐equation exists, Newton’s method may be used to compute the solution by iteration.

Generalized method of a resolvent operator expansion. III
View Description Hide DescriptionWe show that our Born‐like parametrized expansion of R(E)=(E−H)^{−1} is, after minor modifications, well defined even at the pole of R(E) and provides a new method of solving linear homogeneous equations. Three different possibilities of application are discussed here: (1) An analytic method of solving the differential eauations. It is based on the partitioning of generalized power series and illustrated by the new solution of the s‐wave Schrödinger equation. (2) A consequent model space reduction of Schrödinger equation. In terms of the matrix moments of H, the effective interaction is defined as an operator continued fraction. (3) A new form of the perturbation theory which dispenses with the solution of the unperturbed problem.

Regularization of Hamiltonians and processes
View Description Hide DescriptionFor Markov processes and Hamiltonians given by energy forms, singular drift coefficients and potentials, more general than distributions, are allowed. We show that the forms can be regularized in such a way that the approximating processes and semigroups converge as the regularization is removed.

Absorption time by a random trap distribution
View Description Hide DescriptionConsider a particle performing a random walk to nearest neighbors on a simple cubic lattice in any number of dimensions D. The lattice contains a fraction q of randomly located ’’trapping’’ sites which absorb the walker when stepped on. We calculate the mean time to trapping. This involves the expected number V(t) of distinct sites that a walker would visit in t steps in absence of traps, a quantity known only asymptotically for large t; however, an exact calculation possible in one dimension suggests that the results are unexpectedly precise. The trapping time is proportional to q ^{−1} for D=3 or greater, and more complicated for D=1,2.

Unidirectional wave propagation in one‐dimensional first‐order Hamiltonian systems
View Description Hide DescriptionDefining the velocity of a conserved density to be the velocity of the center of gravity of this density, it is shown that for linear equations this velocity equals the weighted group velocity (with the density as weight function). For nonlinear equations, expressions for the centrovelocity of several conserved densities are derived. In particular, for a class of nonlocal equations, the centrovelocity of the energy density turns out to be some weighted average of the group velocity of the corresponding linearized equation. For a specific equation of this type, viz, the BBM equation, it is shown that upon restricting it to solutions whose initial form represents a long, low wave, the centrovelocity of the energy density is positive for all positive time.

Symmetries and vacuum Maxwell’s equations
View Description Hide DescriptionA new class of infinitesimal symmetries are given for Maxwell’sequations in the absence of sources.

Para‐Fermi quantization in the representation of SO(n)
View Description Hide DescriptionThe result of the representation theory of SO(n) is applied to the quantization of f para‐Fermi oscillators, and it is shown that a special irreducible representation of SO(2f+1) is realized in the quantization of the field with order p when the condition of vacuum is used. The group theoretical meaning of the quantities such as vacuum, order p, and the space of the state in the para‐Fermi quantization is made clear, and it is seen that the space of the state vectors corresponds to that of the single‐ or double‐valued representations according to p even or odd.

High‐energy behavior of renormalized Feynman amplitudes. II
View Description Hide DescriptionThe high‐energy polynomial and logarithmic behavior of renormalized Feynman amplitudes, involving subtractions, is studied with total generality when some or all of the external momenta of the graphs in question become large in Euclidean space nonexceptionally for theories which on experimental grounds may contain z e r o mass particles. Initially, all the propagators are defined to have nonzero masses in their denominators. The vanishing masses are led to vanish, and in general, at different rates. Rules are given for applications and examples are then worked out to demonstrate the application of these rules.

Simplified Bose description of para‐Bose operators
View Description Hide DescriptionA set of n para‐Bose operators of order p is explicitly constructed in terms of n×p Bose operators.

Quadratic Hamiltonians in phase space and their eigenstates
View Description Hide DescriptionThe solution of the Schrödinger equation for a Hamiltonian H that is a general second order polynomial in the canonical coordinates q _{ i } and momenta p _{ i } is discussed. Examples of such Hamiltonians abound in various fields of physics and, e.g., the problem of small vibrations has been solved a long time ago. In the general case we first use linear canonical transformations [the group Sp (2n, R)] to reduce H to a simplified representative for H_{ R }. Next we imbed each H_{ R } into a complete set of commuting second order integrals of motion, making use of a recently obtained classification of maximal Abelian subalgebras of the algebra sp (2n, R). This imbedding is then used to separate variables in the representative Schrödinger equations and to obtain complete sets of their eigenfunctions. Finally the solutions of the representative equations can be transformed back into those of the original one making use of representations of the canonical transformations. The program is implemented fully for n=1,2 and its structure is analyzed for arbitrary n. Special cases of interest are treated in detail, e.g., Hamiltonians for a system of n particles invariant under rotations, translations and permutations.