Volume 23, Issue 4, April 1982
Index of content:

Explicit expressions for double Gel’fand states
View Description Hide DescriptionDouble Gel’fand states are given explicitly as polynomials of boson operators. The coefficients of these polynomials may be expressed in terms of recoupling coefficients of external products of permutation groups or Kronecker products within the unitary group, as well as in terms of multiple Clebsch–Gordan coefficients of the unitary group.

Geometric quantization and UIR’s of semisimple Lie groups. II. Discrete series
View Description Hide DescriptionThe Auslander–Kostant induction scheme is extended to yield the discrete series UIR’s of G (G semisimple) in the L ^{2}‐cohomology group H^{ i } _{0} (L_{λ}), where i = (1/2)(dimk−rankg) or i = (1/2)dim p. We show that this involves a choice of complex structure in each Weyl chamber which is intimately connected with the choice of a positive polarization at that point. We illustrate our results with the example of Spin (4,2).

Geometric quantization and UIR’s of semisimple Lie groups. III. Principal and supplementary series
View Description Hide DescriptionThe principal and supplementary series representations of arbitrary semisimple Lie groups is analyzed in the framework of Auslander and Kostant’s theory of UIR’s of solvable Lie groups. As an illustration, we discuss a physically relevant symmetry group, Spin (4,2).

Anomalies and eigenvalues of Casimir operators for Lie groups and supergroups
View Description Hide DescriptionAn expression for the anomaly of any representation of the SU(N) groups and SU(N/M) supergroups is presented. Some anomaly free complex representations of SU(N) and SU(N/M) are pointed out. For SU(N) these occur for N?5 and for large dimensions. For SU(N/M) they already occur for low dimensions. Also a generating function to obtain the eigenvalues of all Casimir operators of SU(N/M) is given and an extension to other supergroups is pointed out.

Dimensions of orbits and strata in complex and real classical Lie algebras
View Description Hide DescriptionExplicit expressions for codimensions of orbits and strata of elements of complex Lie algebrasg l(n, C), o(n, C), s p(2n, C) and real Lie algebrasg l(n, R), g l(n, H), u( p, q), o( p, q), o*(2n), s p(2n, R), and s p(2p, 2q) are given. They make it possible to list easily all dimensions of orbits and strata in a given Lie algebra. The dimension of an orbit or stratum of a given matrix, an element of one of the Lie algebras in its natural representation, can be determined from our formulas after the matrix has been transformed into its Jordan normal form in g l(n, C), g l(n, R), or g l(n, H). Stratification of the Lie algebra in the vicinity of a singular element is discussed.

Application of a new functional expansion to the cubic anharmonic oscillator
View Description Hide DescriptionA new representation of causal functionals is introduced which makes use of noncommutative generating power series and iterated integrals. This technique allows the solutions of nonlinear differential equations with forcing terms to be obtained in a simple and natural way. It generalizes some properties of Fourier and Laplace transforms to nonlinear systems and leads to effective computations of various perturbative expansions. Illustrations by means of the cubic anharmonic oscillator are given in both the deterministic and the stochastic cases.

Nonlinear superposition, higher‐order nonlinear equations, and classical linear invariants
View Description Hide DescriptionWe find a class of nonlinear ordinary differential equations, in ρ, of any order m?2 whose solutions are given by the nonlinear superposition law ρ(t) = x(t)r(τ), dτ = μ(t)d t, where d ^{ m } r/dτ^{ m } = F(r, r ′,...) and x(t) satisfies a special self‐adjoint linear equation of order m. The coefficients of the self‐adjoint equation in x, which are identical to those of the nonlinear equations in ρ, can be deduced from the well known invariants of the classical theory of linear differential equations.

On one method of solving the Helmholtz equation
View Description Hide DescriptionThis paper, Paper II of a series, presents a transformation technique that enables one to substantially simplify the form of the matrix elements of the simultaneous equations to which the two‐dimensional Helmholtz equation was reduced in Paper I. The explicit passage to the limit was also carried out for the case of the normally incident wave.

Pseudopotentials and Lie symmetries for the generalized nonlinear Schrödinger equation
View Description Hide DescriptionThe generalized nonlinear Schrödinger equationz _{ x x }+i z _{ t } = f(z, z*) is analyzed from the point of view of the existence of pseudopotentials, Bäcklund transformations, Lie symmetries, and conservation laws. Applying the Wahlquist–Estabrook method of closed differential ideals we show that eight classes of nontrivial interaction terms f(z,z*) exist for which the equation allows the existence of pseudopotentials. Five of them simply lead to conservation laws, the remaining three to Bäcklund transformations. The usual ’’cubic’’ nonlinear Schrödinger equation with f(z,z*) = εz‖z‖^{2} is obtained as a special case. It is also the only case for which the Bäcklund transformation contains a free parameter. We show that the real and complex parts of this parameter are generated by the dilation and Galilei invariance of the equation.

Exact recursion relation for pseudobidimensional arrays of dumbbells
View Description Hide DescriptionExact relationships are developed that describe the occupation statistics for pseudobidimensional arrays of dumbbells. It is found that E(k,N), the number of ways of arranging k indistinguishable dumbbells on a 3×N diagonal array of compartments is exactly described by the recursion relation E(k,N) = 4E(k−1,N−1)+E(k,N−1)−2E(k−2,N−2). The 3×N diagonal lattice space provides to the central sites of the lattice their full coordination number of nearest‐neighboring compartments; therefore the solution of the present system provides a 1×N diagonal (central sites) ’’window’’ to watch the behavior of a true bidimensional array of dumbbells. For large values of N, the dumbbell freedom per lattice site on any given site of the central diagonal is found to be 1.8477⋅⋅⋅, which is only 3% higher than the exact value (1.7916⋅⋅⋅) found for a ’’locked’’ dumbbell on a bidimensional array.

Analytic structure of the Henon–Heiles Hamiltonian in integrable and nonintegrable regimes
View Description Hide DescriptionThe solutions of the Henon–Heiles Hamiltonian are investigated in the complex time plane. The use of the ’’Painlevé property,’’ i.e., the property that the only movable singularities exhibited by the solution are poles, enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable. Special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes. These boundaries have a remarkable self‐similar structure whose form changes as a function of the nonlinear coupling.

Classical mechanics of nonspherical bodies. I. Binary collisions in two dimensions
View Description Hide DescriptionIn order to discuss the statistical mechanics and to prove an H‐theorem for two‐dimensional, nonspherical bodies, which possess only a symmetry axis, the mechanics of collisions of such bodies are investigated in detail, starting from the collision formulae of D. Bernoulli and Euler.

Classical mechanics of nonspherical bodies. II. Boltzmann equation and H‐theorem in two dimensions
View Description Hide DescriptionAn H‐theorem is proved for a gas of two‐dimensional rigid bodies which are not spherically symmetric but possess only a symmetry axis.

Time‐dependent quadratic constants of motion, symmetries, and orbit equations for classical particle dynamical systems with time‐dependent Kepler potentials
View Description Hide DescriptionIt is shown there are only two classes of time‐dependent Kepler potentials [V _{2}≡λ_{0}(a t ^{2}+b t+c)^{−1/2}/r, (b ^{2}−4a c≠0), and V _{3}≡λ_{0}(αt+β)^{−1}/r] for which the associated classical dynamical equations will admit quadratic first integrals more general than quadratic functions of the angular momentum. In addition to the angular momentum the system defined by V _{2} admits only a ’’generalized time‐dependent energy integral,’’ while the system defined by V _{3} admits in addition to these a time‐dependent vector first integral that is a generalization of the Laplace–Runge–Lenz vector constant of motion (associated with the time‐independent Kepler system). For the V _{3} system the time‐dependent vector first integral is employed to obtain in a simple manner the orbitequations in completely integrated form. The complete group of (velocity‐independent) symmetry mappings is obtained for each of these two classes of dynamical systems and used to show that the generalized energy integral is expressible as a Noether constant of motion.

Direct and inverse scattering problems of the nonlinear intermediate long wave equation
View Description Hide DescriptionThe inverse scattering transformation method associated with a nonlinear singular integrodifferential equation is discussed. The equation describes long internal gravity waves in a stratified fluid of finite depth, and reduces to the Korteweg–de Vries equation as shallow water limit and the Benjamin–Ono equation as deep water limit. Both limits of the method and novel aspects of the theory are also discussed.

Parabolic approximations to the time‐independent elastic wave equation
View Description Hide DescriptionA splitting matrix method is used to derive two parabolic‐approximation partial differential equations to the three‐dimensional, linear, elastic wave equation in isotropic, inhomogeneous media. The derivation is valid for media whose Lamé parameters vary slowly on the length scale of the wavelength of the elastic waves. Next an integral form of the full wave equation is derived based on the splitting matrix and the parabolic approximation solution. Iteration of this equation gives a three‐dimensional vector‐valued series which generalizes the one‐dimensional Bremmer series (which was used in the study of second‐order ordinary differential equations). These results are expected to have applications to geophysical modeling and nondestructive evaluation.

Group representations in the Liouville representation and the algebraic approach
View Description Hide DescriptionGroup representations on the Liouville representation spaces are considered. It is shown that the state space I_{1}(H) of trace class operators on Hilbert spaceH and the observable space L(H) of bounded operators are completely reducible under physically induced representations of compact Hausdorff groups when appropriate topologies are used. For state space I_{1}(H) both the norm topology and the weak topology lead to complete reducibility, while for observable space L(H) the weak‐* topology—but not the norm topology—suffices. This leads to conservation laws, selection rules, and Wigner–Eckart theorems for the Liouville representation. It is shown that serious difficulties are encountered when a similar theory is attempted on the observable space and state space used in the algebraic approach.

Inverse scattering. IV. Three dimensions: generalized Marchenko construction with bound states, and generalized Gel’fand–Levitan equations
View Description Hide DescriptionThis paper represents the final installment in a series on the solution of the inverse scattering problem for the Schrödinger equation in three dimensions. The potential is constructed from a given scattering amplitude without assuming its existence, even in the presence of bound states. For exponentially decreasing potentials, properties of the Jost function and of the regular solution are derived that are sufficient to establish the triangularity of the kernel on which the generalized Gel’fand–Levitan (GL) equation is based. Other generalized GL equations, for nonzero reference potentials, and a nonlinear equation are derived, and for central potentials they are shown to reduce to the well‐known radial equations. The contents of the series of papers is summarized.

Maximum of the spin‐flip cross section from unitarity and three constraints
View Description Hide DescriptionA sharper upper bound on the spin‐flip cross section is found by applying the variational calculus with three equality constraints and unitarity. These are the total cross section, elastic cross section, and the forward slope. Unitarity of the partial waves provides the inequality constraints.

Building things in general relativity
View Description Hide DescriptionIn the context of general relativity, there is a sense in which certain objects cannot be constructed, using reasonable matter, from normal initial conditions. An attempt is made to capture this sense as a definition. The implications of such a definition, along with some related results and open questions, are discussed.