Volume 31, Issue 7, July 1990
Index of content:

Multivariable biorthogonal continuous–discrete Wilson and Racah polynomials
View Description Hide DescriptionSeveral families of multivariable, biorthogonal, partly continuous and partly discrete, Wilson polynomials are presented. These yield limit cases that are purely continuous in some of the variables and purely discrete in the others, or purely discrete in all the variables. The latter are referred to as the multivariable biorthogonal Racah polynomials. Interesting further limit cases include the multivariable biorthogonal Hahn and dual Hahn polynomials.

The unitary irreducible representations of SU(2,1)
View Description Hide DescriptionThis paper analyzes the irreducible unitary representations of SU(2,1) in a basis labeled as ‖p, q; j m y〉, where p,q correspond to quantum numbers associated with the quadratic and cubic Casimir operators, j,m label states of the SU(2) subgroup, and y labels the quantum number with respect to the U(1) subgroup. All the irreducible representations are found and the allowed range of these quantum numbers for each representation are given. The results are expressed in the form of diagrams that show the allowed values in a (j,y) plot for fixed values of p,q. A (p,q) plot is also provided that indicates the allowed values of these quantum numbers.

Symplectic orbits in quantum state space
View Description Hide DescriptionAll symplectic orbits of the action of an arbitrary compact connected Lie group on the space of density operators are found. It is shown that there is only one orbit that is Kähler (orbit of coherent states).

Relationship between S _{ N } and U(n) isoscalar factors and higher‐order U(n) invariants
View Description Hide DescriptionGeneral relationships expressing U(n) and S _{ N } coupling and transformation isoscalar factors in terms of U(n) Racah and 9λ coefficients are derived. The absolute values of U(n) Racah coefficients involving at most k‐column irreducible representations are shown to be identical with SU(k) Racah coefficients. In particular, an explicit relationship is established between the U(n) and SU(2) approaches to the many‐electron correlation problem.

On some Racah coefficients of U(n)
View Description Hide DescriptionSome U(n) Racah coefficients appearing in various applications of vector coherent state and K‐matrix theories are calculated: For such a purpose, use is made of their definition in terms of U(n):U(n−1) reduced Wigner coefficients and U(n−1) Racah coefficients. By starting from known U(2) Racah coefficients, the recursion relations obtained are solved by induction over n.

Remarks on tensor operators
View Description Hide DescriptionThe notion of an algebra of tensor operators for a simple Lie algebra is discussed. A model for the finite‐dimensional irreducible representations of sl(4) is constructed. Explicit Wigner operators acting on the model are defined. Striking commutation properties for these operators are conjectured that resolve a sequence of nontrivial multiplicity problems.

Moment invariants for the Vlasov equation
View Description Hide DescriptionMoment invariants [functions of the moments of a Vlasov distribution that are invariant under Sp(6)] are classified using Young diagrams. The connection between the moment invariants and the Poincaré invariants is established. An application using the moment invariants as phase space coordinates is considered for a matching section in a particle‐beam accelerator, and a Lie–Poisson numerical integration algorithm for the moment dynamics is proposed.

Nonlinear evolution equations related to the first‐order system: φ_{ x } =λJφ+Pφ
View Description Hide DescriptionLet U _{ t } =F(U,∂U/∂x,..., ∂^{ r } U/∂x ^{ r },t) or U _{ x t } =F(U,∂U/∂x,...,∂^{ r } U/∂x ^{ r },t) be the nonlinear evolution equations that are the compatibility conditions between φ_{ x } =λJφ+Pφ and φ_{ t } =Aφ for P=U(x,t) or P=U _{ x }(x,t), respectively. In this paper, it is proved that if A(Z _{0},...,Z _{ r−1},t,λ) is a continuous function such that (∂A/∂Z _{ k }) (Z _{0},...,Z _{ r−1},t,λ) k=0,1,..., exists (Z _{ k }=∂^{ k } U/∂x ^{ k }), then for P=U(x,t), A is a polynomial in λ of degree r and for the case P=U _{ x }, A=A _{−} _{1}/λ+A _{0}+⋅⋅⋅+A _{ r−1}λ^{ r−1}. The case where P=Z _{ m }, m≥2 is also analyzed.

Relations between hyperspherical harmonic transformations and generalized Talmi–Moshinsky transformations
View Description Hide DescriptionThe correlations between the hyperspherical harmonic transformations and the generalized Talmi–Moshinsky transformations are studied for the three‐body and four‐body systems. An optical approach for solving few‐body problems through diagonalizing the Hamiltonian of a system in an optimal subset of the basis functions of harmonic oscillators in hyperspherical coordinates is proposed. The evaluations of the interaction matrix elements are achieved with the aid of the transformation properties of hyperspherical harmonics.

Lie group symmetries and invariants of the Hénon–Heiles equations
View Description Hide DescriptionLie group symmetries and invariants of the generalized Hénon–Heiles equations are found. The coupled second‐order equations are invariant under translations in time, in general, and the stretching group (dilation) if the linear terms in the ‘‘force’’ are absent. The equivalent set of four coupled first‐order equations is found to be invariant under a one‐parameter group for three cases and the group generators are given. Three different approaches are reported: the ‘‘classical method’’ for determining Lie group symmetries, a modified method for finding Lie group symmetries with vector fields and the direct method for calculating the invariants. For the Hénon–Heiles equations the direct method is the most efficient.

Exponentially evolving densities
View Description Hide DescriptionThe notion of a conserved density is generalized to those densities whose integrals evolve exponentially. In the formal differential algebra used to find them, only a slight modification of the conserved density case is needed. Various examples for quasilinear evolution equations are given and an application to finding higher‐order conserved densities by taking the Poisson bracket of exponentially evolving densities is indicated.

Equivariant harmonic maps into homogeneous spaces
View Description Hide DescriptionThis paper is about harmonic maps from closed Riemann surfaces into homogeneous spaces such as flag manifolds and loop groups. It contains the construction of a family of new examples of harmonic maps from T ^{2}=S ^{1}×S ^{1} into F(n) or Ω( U(n)) that are not holomorphic with respect to any almost complex structure on F(n) or Ω( U(n)), where F(n) is the quotient of U(n) by any maximal torus and Ω(u(n)) consists of f: S ^{1}→U(n) smooth such that f(1)=I.

Geometry of linear pairs for self‐dual gauge fields
View Description Hide DescriptionA linear pair for self‐dual gauge fields is constructed for the metric d s ^{2}=g _{ z z̄} d z d z̄+g _{ yy }̄d y d ̄. It is shown that for consistency g _{ z z̄} and g _{ y ȳ}, apart from a possible overall conformal factor, are given in terms of two Liouville fields of equal and opposite curvatures. The null surface corresponding to the pair and the homogeneous solutions, playing a fundamental role, are constructed explicitly. The five‐dimensional space of y,ȳ,z,z̄ and the spectral parameter λ is studied. The proper transformation of λ corresponding to holomorphic ones of y and z is found. Known monopole, instanton, and (quasi)periodic solutions are all shown to emerge systematically as particular cases of our formalism. As examples of new possibilities, the case of accelerated observers and that of cosmic string backgrounds are presented.

Parabose algebras and subgraph polynomials
View Description Hide DescriptionA graph theoretical method is proposed for the calculation of inner products in the Fock spaces of parabose algebras. For this purpose, a new class of polynomials associated with finite graphs is introduced. The obtained results can be generalized to the parafermi case.

Hannay’s angles outside the adiabatic range: Coupled linear oscillators near resonance
View Description Hide DescriptionThe simple harmonic 2‐degree of freedom (dof) oscillator with the potential V= (1)/(2) [(1+r sin εt)q ^{2} _{1}+(1−r sin εt) q ^{2} _{2}+2r cos εt q _{1} q _{2}] is considered. It is shown that a coupling parameter k∼r/ε determines the behavior of the system after a loop 0≤t≤2π/ε in parameter space. Somewhat unexpected mode conversion and phase corrections occur. An optical model is outlined.

The linearized gauge field propagator in an inhomogeneous medium
View Description Hide DescriptionIn an inhomogeneous medium with μ=κ^{−} ^{1}, the linearized gauge field propagator is presented in terms of spherical harmonics for the Coulomb gauge. Explicit expressions for the propagator are given in the case of the MIT bag for both the Coulomb and Lorentz gauges. Gauge invariance is shown explicitly for the one gluon interaction in the MIT bag. This work supplements and corrects the earlier work by Bickeböller, Goldflam, and Wilets [J. Math. Phys. 2 6, 1810 (1985)].

Refined asymptotic expansion for the partition function of unbounded quantum billiards
View Description Hide DescriptionThis paper presents a refined asymptotic expansion for the partition function Θ(t)=Tr e ^{ tΔ} of quantum billiards in the unbounded regions {0≤x,0≤y x ^{μ}≤1}, μ>0, and {0≤y e ^{‖x‖}≤1}⊆R^{2}, where Δ is the Dirichlet Laplacian. Simon [Ann. Phys. 1 4 6, 209 (1983); J. Funct. Anal. 5 3, 84 (1983)] determined the leading divergence of the trace of the heat kernel for the first class of systems. Standard techniques are combined for the evaluation of Θ for bounded region billiards with results by Van den Berg [J. Funct. Anal. 7 1, 279 (1987)] for ‘‘horn‐shaped regions’’ using an optimized way of dividing the region into ‘‘narrow’’ and ‘‘wide’’ parts to determine the first three terms in the asymptotic expansion of Θ. Results are also stated for bounded regions with cusps that can be obtained by the same method. As an application, the spectral staircase of the strongly chaotic billiard system defined in the region {0≤x y≤2,x≥0}, which has been discussed in connection with the Riemann ζ function and the search for quantum chaos is considered.

Exact solution of the n‐dimensional Dirac–Coulomb equation
View Description Hide DescriptionAn exact solution of the n‐dimensional Dirac–Coulomb equation is obtained with the radial wave function containing only one term of a confluent hypergeometric function. It is of the same form as the solutions to the Schrödinger and Klein–Gordon equations with a Coulomb potential in n dimensions.

Extended Hilbert space approach to few‐body problems
View Description Hide DescriptionA general formulation of the quantum scattering theory for a system of few particles, which have an internal structure, is given. Due to freezing out the internal degrees of freedom in the external channels, a certain class of energy‐dependent potentials is generated. By means of potential theory, a modified Faddeev equation is derived both in external and internal channels. The Fredholmity of these equations is proven and this is what provides a sound basis for solving the addressed scattering problem.

Low‐frequency moments in inverse scattering theory
View Description Hide DescriptionThe low‐frequency moments of the scattering amplitude are utilized in order to identify the capacity, the center, and the orientation of an acoustically soft scatterer.