Index of content:
Volume 33, Issue 6, June 1992

Representations of the quantum algebras U_{ q }(u _{ r,s }) and U_{ q }(u _{ r+s }) related to the quantum hyperboloid and sphere
View Description Hide DescriptionRepresentations π_{Λμ} of the principal degenerate series of the quantum algebraU_{ q }(u _{ r,s }) are introduced. Structure of these representations is studied. Several classes of unitary irreducible representations of the algebraU_{ q }(u _{ r,s }) are separated in the set of irreducible representations π_{Λμ} and irreducible constituents of reducible representations π_{Λμ}. The formulas of action of operators of irreducible representations of the quantum algebraU_{ q }(u _{ r+s }) in the basis corresponding to restriction of representations onto the subalgebra U_{ q }(u _{ r }+u _{ s }) are given.

Computation of error effects in nonlinear Hamiltonian systems using Lie algebraic methods
View Description Hide DescriptionThere exist Lie algebraic methods for obtaining transfer maps around any given trajectory of a Hamiltonian system. This paper describes an iterative procedure for finding transfer maps around the same trajectory when the Hamiltonian is perturbed by small linear terms. Such terms often result when an actual system deviates from an ideal one due to errors. Two examples from accelerator physics are worked out. Comparisons with numerical computations, and in simple cases exact analytical calculations, demonstrate the validity of the procedure.

Casimir operators, duality, and the point groups
View Description Hide DescriptionIn atomic or nuclear shell theory the quadratic Casimir operators can be used to evaluate the expectation values of two‐body interactions among equivalent members of a shell. Racah has given closed expressions for these Casimir operators for all the groups in the Racah chain. In this paper an alternate derivation of these closed expressions based on duality with the symmetric group is given. The derivation allows extension of these expressions to situations in which equivalent spins are in an environment, such as a point group, for which the two‐body interactions separate into distinct sets. Possible application to NMR systems with spin ≥1/2 are discussed.

Universal L operator and invariants of the quantum supergroup U _{ q } (gl(m/n))
View Description Hide DescriptionA spectral parameter‐dependent solution of the graded Yang–Baxter equation is obtained, which is universal in the sense that it lives in U _{ q } (gl(m/n))⊗End(V) with V the vector module of U _{ q } (gl(m/n)). The invariants of this quantum supergroup are constructed using this solution.

Comment on ‘‘The matrix representation of U _{4} in the U _{2}×U _{2} basis and isoscalar factors for U _{ p+q }⊇U _{ p }×U _{ q }’’ [F. Pan, J. Math. Phys. 31, 1333 (1990)]
View Description Hide DescriptionComplementary group technique leads to more simple solutions of some problems considered by F. Pan [J. Math. Phys. 31, 1333 (1990)], including special resubducing coefficients and isoscalar factors of unitary groups.

Explicit canonical tensor operators and orthonormal coupling coefficients of SU(3)
View Description Hide DescriptionThe canonical unit SU(3) tensor operators are constructed by means of the stretched coupling of the auxiliary maximal and minimal null space tensor operators, with the renormalization factors expressed in terms of the denominator functions of Biedenharn, Gustafson, Lohe, Louck, and Milne. The matrix elements of the maximal null space tensor operators are expressed with the help of the modified projection operators of Asherova and Smirnov. The self‐conjugate minimal null space tensor operators are expressed in terms of the group generators with the help of the weight lowering operator technique. The corresponding extreme isoscalar factors of the Clebsch–Gordan (Wigner) coefficients are used as constructive elements of the explicit recursive expression for the general orthonormal isoscalar factors of SU(3) with its considerable simplication for the boundary values of parameters. The general isofactors are also expanded in the different ways in terms of their boundary values. The new classes of the generalized hypergeometric series are used as constructive elements of the SU(3) and SU(2) representation theory functions and their properties are considered.

Evaluation of the spherical Bessel transform of a Whittaker function: An application of a difference equation method
View Description Hide DescriptionA simple analytic expression for the spherical Bessel transform of the zero‐range bound state wave function with the Coulomb interaction present, for the lth partial wave, expressed as a Whittaker function, is obtained. The result is given in terms of polynomials of degree l, the exponential function, and a simple hypergeometric function which is independent of l. Transformations of this latter function are derived in terms of more rapidly convergent series. The method presented has much wider application, since it relies essentially only on the existence of differential‐difference equations for the functions involved, and the solution of the inhomogeneous difference and differential equations satisfied by the transform.

Cauchy problem for the linearized version of the Generalized Polynomial KdV equation
View Description Hide DescriptionIn the present paper results about the ‘‘Generalized Polynomial Korteweg–de Vries equation’’ (GPKdV) are obtained, extending the ones by Sachs [SIAM J. Math. Anal. 14, 674 (1983)] for the Korteweg–de Vries (KdV) equation. Namely, the evolution of the so‐called ‘‘prolonged squared’’ eigenfunctions of the associated spectral problem according to the linearized GPKdV is proven, the Lax pairs associated with the ‘‘prolonged’’ eigenfunctions as well as ‘‘prolonged squared’’ eigenfunctions are derived, and on the basis of some expansion formulas the Cauchy problem for the linearized GPKdV with a decreasing at infinity initial condition is solved.

A first integral for a class of time‐dependent anharmonic oscillators with multiple anharmonicities
View Description Hide DescriptionThe integrability of the equationq̈=f(t)q ^{2}+g(t)q ^{3}+h(t)q ^{4}+j(t)q ^{5} is considered. Particular cases of this equation arise in the study of charged plasma in an axially symmetric magnetic field and in shear‐free spherically symmetric gravitational fields in general relativity. The above equation with only a quadratic term arises in the study of shear‐free fluids, in general [A. Krasinski, J. Math. Phys. 30, 433 (1989)]. The equation with a cubic term is applicable when there is an electromagnetic field. In special cases we reduce the solution to a quadrature that has solutions in terms of elliptic integrals. A Lie point symmetry analysis is performed and the different cases that arise are considered for the existence of a symmetry.

Schlesinger transformations of Painlevé II‐V
View Description Hide DescriptionThe explicit form of the Schlesinger transformations for the second, third, fourth, and fifth Painlevé equations is given.

P determinants and boundary values
View Description Hide DescriptionIt is shown that a regularized determinant based on Hilbert’s approach (which we call the ‘‘p determinant’’) of a quotient of elliptic operators defined on a manifold with boundary is equal to the ‘‘p determinant’’ of a quotient of pseudodifferential operators. The last ones are entirely expressible in terms of boundary values of solutions of the original differential operators. It is argued that, in the context of quantum field theory, these boundary values also determine the subtractions (i.e., the counterterms) to which this regularization scheme gives rise.

The supertrace of the steady asymptotic of the spinorial heat kernel
View Description Hide DescriptionAn essentially self‐contained, rigorous proof of the Atiyah–Singer index theorem is given for the case of the twisted Dirac operator. Indeed, the stronger local index theorem, which implies the general case, is proven here by computing the supertrace of the steady (time‐independent) asymptotic of the twisted spinorial heat kernel. The computation is carried out in the spirit of Patodi’s proof of the Gauss–Bonnet–Chern theorem. Gilkey’s approach involving invariant theory is not used, but rather a generalization of Mehler’s formula is derived and utilized along with some elementary properties of Clifford algebras. The use of Mehler’s formula was inspired by work of Getzler, but families of Clifford algebra‐valued pseudodifferential operators, and limits and estimates of families of heat kernels, are avoided here. Asymptotic expansions of heat kernels for operators on Euclidean fields are of fundamental importance in the computation of quantum corrections to the effective action for classical fields. Thus, the general formula developed here for the asymptotics may also be helpful in this regard.

A relation between group‐ and Čech‐cohomology in principal fiber bundles and anomalies
View Description Hide DescriptionA natural map between group‐cohomology of the structure group of a principal fiber bundle with coefficients in the space of functions from the total space into an Abelian group and Čech‐cohomology of the base space is defined. A differential complex of local group‐cochains is constructed and an analog of the Poincaré lemma for group‐cohomology is proven. By using the machinery of spectral sequences the cohomology of this complex is calculated and the connection between group‐cohomology and Čech‐cohomology of the given principal fiber bundle is elucidated. Finally, the non‐Abelian and Witten anomaly in this context is reviewed and the relevance of our results for lifting principal group actions is discussed.

Continuous integrable systems with multi‐Poisson brackets
View Description Hide DescriptionFor discrete finite systems, it is known that the zero Nijenhuis tensor condition can be used to explicitly construct conserved quantities in involution. Although the same method is not directly generalizable for continuous systems because of the divergence problem, the difficulty can be overcome with some minor modifications. In this way, various cases of KdV, nonlocal KdV, continuous Toda lattice, Kac–Moody hierarchy, and a model based upon a W _{3}‐algebra systematically by the same technique can be discussed.

Neutrino in the presence of gravitational fields: Separation of variables
View Description Hide DescriptionIt is well known that the most complete information about single‐particle states is contained in its wave function. For spin‐1/2 particles this means that it is necessary to have exact solutions of the Dirac equation. In particular, in the case of neutrinos in the presence of gravity, it is necessary to solve the covariant Dirac equation. At present, the existence of neither massless neutrinos(electron neutrinos) nor massive neutrinos(muon and τ neutrinos) cannot be excluded. Since for massless neutrinos any solution of the Dirac equation is also a solution of the Weyl equation, there exists the possibility of studying, from a unified point of view, massive as well as massless neutrinos by means of the Dirac equation. In the search of exact solutions of systems of partial differential equations one can proceed as follows: (a) separation of variables and (b) solution of the corresponding ordinary differential equations. In the present paper, a complete analysis of the separation of variables in the Dirac equation for massive as well as for massless neutrinos is carried out by means of the algebraic method [J. Math. Phys. 30, 2132 (1989)]. It is found that for the massless neutrinos, there are further possibilities of separation of variables, not valid for the massive case.

Dixon–Selberg summation over local number fields
View Description Hide DescriptionThe character summation formula analogous to Dixon’s theorem for the hypergeometric function _{3} F _{2} is derived for R and Q _{ p }.

How to construct finite‐dimensional bi‐Hamiltonian systems from soliton equations: Jacobi integrable potentials
View Description Hide DescriptionA systematic method of constructing finite‐dimensional integrable systems starting from a bi‐Hamiltonian hierarchy of soliton equations is introduced. The existence of two Hamiltonian structures of the hierarchy leads to a bi‐Hamiltonian formulation of the resulting finite‐dimensional systems. The case of coupled KdV hierarchies is studied in detail. A surprising connection with separable Jacobi potentials is uncovered and described.

Compact classical systems: so(n) systems
View Description Hide DescriptionCompact quantum systems based on compact kinematical Lie algebras have been described previously [J. Math. Phys. 29, 1521 (1988)]. A ‘‘compact classical system’’ is obtained as the classical limit of such a compact quantum system. In this paper, compact classical systems obtained in this way from compact quantum systems based on the special orthogonal Lie algebras so(n) are considered; in particular, those based on so(3) and so(4). Such compact classical systems are shown to exhibit behavior strikingly different from that of corresponding noncompact systems in ordinary classical Hamiltonian dynamics.

Integrals, invariant manifolds, and degeneracy for central force problems in R ^{ n }
View Description Hide DescriptionSince the notion of angular momentum is defined in any dimension by using the exterior product in R ^{ n }, one would guess that central force problems in any dimension are completely integrable, as it is known for n=2 or 3. This is proved explicitly in this paper, by constructing n first integrals independent and involution: the energy and some combinations of the angular momentum components. It is shown that these problems are always reduced to a two‐dimensional plane, and the invariant manifolds are topologically, the Cartesian product of those of the reduced problem times n−2 circle factors. It is also proved that for n≳2 these problems are degenerate in the sense that their Hamiltonians do not satisfy one of the hypotheses of the Kolmogorov–Arnold–Moser theorem for persistence of invariant tori, when one considers small perturbations.

Existence of gauge field in any partially integrable systems
View Description Hide DescriptionFor any integrable and, more generally, any partially integrable model, the existence is shown of a Yang–Mills gauge field with zero curvature, whose gauge group consists of all general coordinate transformations. It has been suggested that this fact may have some connection with existence of an affine connection in the space with zero Riemann curvature tensor. A BRST‐like quadratically nilpotent operator is also constructed for any theory with Poisson bracket structure whenever such a connection with zero Riemann curvature exists.