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  • American Institute of Physics (AIP)  (46,193)
  • Blackwell Publishing Ltd  (37,200)
  • Oxford University Press  (29,994)
  • 1990-1994  (113,387)
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  • 1
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    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 4439-4450 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: An algebraic procedure to compute the higher order symbols of the Heisenberg quantum observables is explicitly implemented. As an application, it is proven in a very simple way that a time evolution under which the symbol of any Heisenberg quantum observable is exactly given by the evolution along the classical flow is necessarily generated by a quadratic Hamiltonian.
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  • 2
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    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 4534-4546 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: It is well known that quantum physics forbids the simultaneous measurement of two noncommuting projection operators. However, sometimes there exists a third projection (mirror projection) which may be measured together with the second one and whose outcomes are completely correlated, in the Einstein–Podolsky–Rosen sense, with the outcomes of the first projection. In this article the states for which such a mirror projection exists are characterized and a procedure for constructing the mirror is given. In particular, it is shown that for two noncommuting projections there are always states for which the mirror projection exists, but also states for which it does not exist.
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  • 3
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    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 4725-4738 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: In this paper all the potential constraints (with or without first-order partial derivatives) of the KP system, from which the associated linear problem can be restricted into a (1+1)-dimensional Hamiltonian equation, are obtained by using the sufficient and necessary condition for a nonlinear equation to be a Hamiltonian system. Some well-known integrable systems, such as (1+1)-dimensional AKNS system, generalized NS system, and several new Hamiltonian equations, are deduced as particular examples.
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  • 4
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    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 4739-4745 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A modified Korteweg–de Vries (mKdV)-type bilinear equation that has been shown to pass the three-soliton solution and the Painlevé tests [see, J. Math. Phys. 2094, 28 (1987); 2572, 31 (1990)] is considered herein. In this article, a Bäcklund transformation and a nonlinear superposition formula are proven rigorously. As an application of the obtained results, N-soliton solutions are obtained. Therefore, the integrability of this mKdV-type bilinear equation is confirmed.
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  • 5
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    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 4799-4808 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A more simple form than the one given by Vaklev [J. Math. Phys. 33, 4111 (1992)] for the soliton solutions related to the quadratic bundle [E. A. Kuznetzov and A. V. Mikhailov, Teor. Mat. Fiz. 30, 303 (1977); D. J. Kaup and A. C. Newell, J. Math. Phys. 19, 798 (1978); V. S. Gerdjikov, M. I. Ivanov, and P. P. Kulish, Teor. Mat. Fiz. 44, 342 (1980); V. S. Gerdjikov and M. I. Ivanov, Bulg. J. Phys. 10, 13 (1983); 10, 2, 129 (1983)] is obtained. The corresponding solutions for the gauge equivalent case are obtained as well. Among them one can find the N-soliton solutions for the derivative nonlinear Schrödinger equation, their gauge equivalents, etc. The method used in the article allows one to express some integrals by means of appropriate determinants and thus the result is obtained without integration.
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  • 6
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    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 6838-6856 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Finite-dimensional Lie algebras of polynomial vector fields on Rn, that contain the elements ∂/∂xi and xi(∂/∂xi) for i=1...n were studied. To any Lie algebra @FL of this class, an N-valued n×n matrix A and a set of special elements S⊆{1,...,n} are associated. It is proven that the pair (A,S) necessarily satisfies two properties. Conversely, to any pair (A,S) satisfying those two properties is associated a Lie algebra @FL(A,S), such that @FL(A,S) is maximal in the class of all @FL with matrix A and special elements S. For the Lie algebras @FL(A,S) the possible extensions to first order differential operators, and its modules of C∞ functions are discussed. © 1994 American Institute of Physics.
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  • 7
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    Journal of Mathematical Physics 35 (1994), S. 4372-4382 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The power density estimation problem with a priori information about spectrum is considered. The problem is reduced to a power density estimation problem with spectral gaps. The solution of the problem is studied in the context of the trigonometric moment problem with gaps. The necessary and sufficient conditions are established for the existence of solution and all solutions are described in indeterminate cases. A new definition of the entropy functional is given. The effective algorithm to construct maximum entropy solution is proposed. A number of explicit illustrations are presented.
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  • 8
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    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 4334-4356 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Standard energy methods are used to study the relation between the solutions of one parameter families of hyperbolic systems of equations describing heat propagation near their parabolic limits, which for these cases are the usual diffusive heat equation. In the linear case it is proven that given any solution to the hyperbolic equations there is always a solution to the diffusion equation which after a short time stays very close to it for all times. The separation between these solutions depends on the square of the ratio between the assumed very short decay time appearing in Cattaneo's relation and the usual characteristic smoothing time (initial data dependent) of the limiting diffusive equation. The techniques used in the linear case can be readily used for nonlinear equations. As an example we consider the theories of heat propagation introduced by Coleman, Fabrizio, and Owen, and prove that near a solution to the limiting diffusive equation there is always a solution to the nonlinear hyperbolic equations for a time which usually is much longer than the decay time of the corresponding Cattaneo relation. An alternative derivation of the heat theories of divergence type, which are consistent with thermodynamic principles, is given as an appendix.
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  • 9
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    Journal of Mathematical Physics 35 (1994), S. 5749-5764 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The Meinardus theorem provides an asymptotic value for the infinite-product generating functions of many different partition functions, and thereby asymptotic values for these partition functions themselves. Under assumptions somewhat stronger than those of Meinardus a more specific version of the theorem is derived, then this is generalized from the spectrum of integers considered by Meinardus to infinite products constructed from an arbitrary spectrum. Included are infinite products that generate partition functions able to count states in the quantum theory of extended objects. There emerges a quite general and explicit method (of which only limited details are given here) for deriving the asymptotic growth of the state density function of multidimensional quantum objects. One finds a universal geometry-independent leading behavior, with nonleading geometry-dependent corrections that the method is capable of providing explicitly.
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  • 10
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    Journal of Mathematical Physics 35 (1994), S. 5844-5849 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Identification of the hierarchy of Calogero–Moser dynamical systems with the dynamical systems of poles of certain solutions to the KP hierarchy is achieved.
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  • 11
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    Journal of Mathematical Physics 35 (1994), S. 5862-5868 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: An explicit construction of the BRST symmetry is presented in the Hamiltonian approach. The construction is based on the splitting homotopy and the transference problem.
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  • 12
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    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 5922-5929 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The divergence theorem in its usual form applies only to suitably smooth vector fields. For vector fields which are merely piecewise smooth, as is natural at a boundary between regions with different physical properties, one must patch together the divergence theorem applied separately in each region. We give an elegant derivation of the resulting patchwork divergence theorem which is independent of the metric signature in either region, and which is thus valid if the signature changes.
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  • 13
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    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 5930-5944 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A general discussion of affine vector fields (collineations) in space–time is given following a previous article by Hall and da Costa. Some general remarks are presented on the zeros of affine vector fields and then the general theory of such zeros is established. Some preliminary results regarding homothetic symmetries in two- and three-dimensional space–times are also given. The impossibility of reducing an affine algebra containing proper affine vector fields to homothetic or Killing vector fields, by means of a change of compatible metric, is demonstrated.
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  • 14
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    Journal of Mathematical Physics 35 (1994), S. 5969-5989 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: This work extends the ideas presented in Taub I [Phys. Rev. D 47, 474 (1993)]. Taub numbers are studied as a set of tensorial conservation laws derivable from curves of solutions to the Einstein equations. The zeroth-order Taub numbers for mass and angular momentum are computed using the Penrose–Goldberg superpotential [Phys. Rev. D 41, 410 (1990)]. A superpotential for the higher-order Taub vectors is given and is gauge invariant with respect to the perturbation gauge. Taub numbers are computed for the Schwarzschild and Kerr solutions, Schwarzschild and Kerr viewed as perturbations of Minkowski and Schwarzschild, respectively, and for the stationary perturbations of Schwarzschild. Taub numbers are found to give a tensorial formulation for mass and angular momentum for asymptotically flat spacetimes. A new solution generation technique for asymptotically flat solutions is also presented.
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  • 15
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    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 6076-6095 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A derivation of the Cesàro–Fedorov relation from the Selberg trace formula on an orbifolded 2-sphere is elaborated and extended to higher dimensions using the known heat-kernel coefficients for manifolds with piecewise-smooth boundaries. Several results are obtained that relate the coefficients, bi, in the Shephard–Todd polynomial to the geometry of the fundamental domain. For the 3-sphere, it is shown that b4 is given by the ratio of the volume of the fundamental tetrahedron to its Schläfli reciprocal.
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  • 16
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    Journal of Mathematical Physics 35 (1994), S. 6123-6143 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The orthogonality relations for SUq(2) are here formulated with the aid of the Woronowicz integral. The first part of this paper is devoted to a summary of the relevant parts of the Woronowicz theory and the second part to its implementation in terms of completely explicit representations of the irreducible unitary representations. The purpose of this presentation is to provide background for a separate study of the q deformation of the Coulomb problem.
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  • 17
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    Journal of Mathematical Physics 35 (1994), S. 6150-6157 
    ISSN: 1089-7658
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    Topics: Mathematics , Physics
    Notes: Finite-dimensional irreducible representations of a reflection algebra related to the reflection equation for the su(1,1)q model are classified. It is proven that this algebra has two inequivalent one-dimensional and one two-dimensional irreducible representations. Some indecomposable representations are also constructed and the (q-boson–)fermion realizations are explicitly presented. Finally, the Yang–Baxterization of this algebra and its constant solutions (the one-dimensional representations) is investigated.
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  • 18
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    Journal of Mathematical Physics 35 (1994), S. 6213-6223 
    ISSN: 1089-7658
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    Topics: Mathematics , Physics
    Notes: A Yang–Baxter operator was canonically associated to any cocommutative braided Lie algebra as defined by Majid. Applying previous results, we construct quantum Koszul complexes for braided Lie algebras with respect to this operator. In the usual case, the complexes reduce to the Chevalley–Eilenberg complexes for classical Lie algebras.
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  • 19
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    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 6226-6226 
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  • 20
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    Journal of Mathematical Physics 35 (1994), S. 6179-6198 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The asymptotic expansions for the Kummer function obtained in the study of the linear response of magnetized Bose plasmas at T=0 K are presented for large and small values of its parameter, thereby displaying the function's asymptotic nonuniformity. The large parameter expansion plays a determining role in the behavior of these Bose systems in the limit that the external magnetic field B→0. This particular expansion is generalized herein and its validity tested by determining the asymptotic expansion for the Hurwitz zeta function.
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  • 21
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    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 6224-6225 
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  • 22
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    Journal of Mathematical Physics 35 (1994), S. 5136-5154 
    ISSN: 1089-7658
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    Topics: Mathematics , Physics
    Notes: A new combinatorial method of constructing four-dimensional topological quantum field theories is proposed. The method uses a new type of algebraic structure called a Hopf category. The construction of a family of Hopf categories related to the quantum groups and their canonical bases is also outlined.
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  • 23
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    Journal of Mathematical Physics 35 (1994), S. 5219-5246 
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    Notes: Feynman diagrams are used to prove a formula for the Jones polynomial of a link derived recently by Reshetikhin. This formula presents the colored Jones polynomial as an integral over the coadjoint orbits corresponding to the representations assigned to the link components. The large k limit of the integral can be calculated with the help of the stationary phase approximation. The Feynman rules allow one to express the phase in terms of integrals over the manifold and the link components. Its stationary points correspond to flat connections in the link complement. A relation between the dominant part of the phase and Milnor's linking numbers is conjectured. It is checked explicitly for the triple and quartic numbers by comparing their expression through the Massey product with Feynman diagram integrals.
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  • 24
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    Journal of Mathematical Physics 35 (1994), S. 6049-6057 
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    Notes: The vector system of linear differential equations for a field with arbitrary fractional spin is proposed using infinite-dimensional half-bounded unitary representations of the SL(2,R) group. In the case of (2j+1)-dimensional nonunitary representations of that group, 0〈2j∈Z, they are transformed into equations for spin-j fields. A local gauge symmetry associated to the vector system of equations is identified and the simplest gauge invariant field action, leading to these equations, is constructed.
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  • 25
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    Journal of Mathematical Physics 35 (1994), S. 6036-6048 
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    Notes: A number of problems in solid state physics and materials science can be resolved by the evaluation of certain lattice sums (sums over all sites of an infinite perfect lattice of some potential energy function). One classical example, the calculation of lattice sums of circular and spherical harmonics, dates back to the last century, to Lord Rayleigh's work on computing the effective conductivity of a simple composite. While Lord Rayleigh presented an efficient asymptotic method for two-dimensional problems, he resorted to direct evaluation of the lattice sums in the three-dimensional case. More recent methods, based on Ewald summation, have been developed by Nijboer and De Wette, Schmidt and Lee, and others. In this article, a fast method for evaluating lattice sums which is based on a new renormalization identity is described.
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  • 26
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    Journal of Mathematical Physics 35 (1994), S. 6158-6171 
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    Notes: For the Z/nZ symmetric elliptic R matrix, L operators which act on the space of A(1)n−1 integrable characters, or the affine Weyl group invariants, of the fixed level l, are constructed. For l=1, the matrix elements of the L operator and those of the R matrix are shown to be the same.
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  • 27
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    Journal of Mathematical Physics 35 (1994), S. 6172-6178 
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    Notes: Lie consistency is defined for commutator expansions and it is shown that while a weakly Lie-consistent q-Campbell–Baker–Hausdorff expansion for a product of two M-type (but not P-type) q exponentials can be formulated, a strongly Lie-consistent expansion only exists for the undeformed case, q=1.
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  • 28
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    Journal of Mathematical Physics 35 (1994), S. 3410-3447 
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    Notes: The aim of this paper is to develop a constraint algorithm for time-dependent Lagrangian systems which permit us to solve the motion equations. This algorithm extends the Gotay and Nester algorithm for autonomous Lagrangians which is, in fact, a particular case. To do this the almost stable tangent geometry of the evolution space and the notion of cosymplectic structure are used.
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  • 29
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    Journal of Mathematical Physics 35 (1994), S. 5101-5135 
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    Notes: By exploiting standard facts about N=1 and N=2 supersymmetric Yang–Mills theory, the Donaldson invariants of four-manifolds that admit a Kahler metric can be computed. The results are in agreement with available mathematical computations, and provide a powerful check on the standard claims about supersymmetric Yang–Mills theory.
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    Journal of Mathematical Physics 35 (1994), S. 5195-5218 
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    Notes: A general definition of Chern–Simons actions in noncommutative geometry is proposed and illustrated in several examples. These examples are based on "space–times'' which are products of even-dimensional, Riemannian spin manifolds by a discrete (two-point) set. If the * algebras of operators describing the noncommutative spaces are generated by functions over such "space–times'' with values in certain Clifford algebras the Chern–Simons actions turn out to be the actions of topological gravity on the even-dimensional spin manifolds. By constraining the space of field configurations in these examples in an appropriate manner one is able to extract dynamical actions from Chern–Simons actions.
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    Journal of Mathematical Physics 35 (1994), S. 5288-5305 
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    Notes: The correspondence of the braid group on a handlebody of arbitrary genus to the algebra of Yang–Baxter and extended reflection equation operators is shown. Representations of the infinite-dimensional extended reflection equation algebra in terms of direct products of quantum algebra generators are derived; they lead to a representation of this braid group in terms of R-matrices. Restriction to the reflection equation operators only gives the colored braid group. The reflection equation operators, describing the effect of handles attached to a three-ball, satisfy characteristic equations that give rise to additional skein relations and thereby invariants of links on handlebodies. The origin of the skein relations is explained and they are derived from an adequately adapted handlebody version of the Jones polynomial. Relevance of these results to the construction of link polynomials on closed three-manifolds via Heegard splitting and surgery is indicated.
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    Journal of Mathematical Physics 35 (1994), S. 5314-5320 
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    Notes: A self-contained, genuinely three-dimensional, monodromy-matrix based, nonperturbative, covariant path-integral approach to polynomial invariants of knots and links in the framework of (topological) quantum Chern–Simons field theory is proposed. The idea of "physical'' observables represented by an auxiliary topological quantum-mechanical model in an external gauge field is introduced as a replacement for the limited notion of the Wilson loop. The possibility of using various generalizations of the Chern–Simons action (also higher-dimensional ones) as well as a purely functional language thereby arises. The theory is quantized in the framework of the antibracket-antifield formalism of Batalin and Vilkovisky, which is well adapted to this case. Using Stokes's theorem and formal translational invariance of the path-integral measure we derive a monodromy matrix corresponding to an arbitrary pair of irreducible representations of an arbitrary semisimple Lie group.
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    Journal of Mathematical Physics 35 (1994), S. 5354-5361 
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    Notes: The semiclassical limit of the quantum Yang–Mills partition function on a compact oriented surface is related to the symplectic volume of the moduli space of flat connections, by using an explicit expression for the symplectic form. This gives an independent proof of some recent results of Witten and Forman.
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    Journal of Mathematical Physics 35 (1994), S. 5494-5506 
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    Notes: The essential role played by differentiable structures in physics is reviewed in light of recent mathematical discoveries that topologically trivial space–time models, especially the simplest one, R4, possess a rich multiplicity of such structures, no two of which are diffeomorphic to each other and thus to the standard one. This means that physics has available to it a new panoply of structures for space–time models. These can be thought of as a source of new global, but not properly topological, features. This article reviews some background differential topology together with a discussion of the role which a differentiable structure necessarily plays in the statement of any physical theory, recalling that diffeomorphisms are at the heart of the principle of general relativity. Some of the history of the discovery of exotic, i.e., nonstandard, differentiable structures is reviewed. Some new results suggesting the spatial localization of such exotic structures are described and speculations are made on the possible opportunities that such structures present for the further development of physical theories.
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    Journal of Mathematical Physics 35 (1994), S. 113-126 
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    Notes: The members Hω=p2+Vω(x) of ergodic families of random operators, defined in H=L2(Rd), are considered as acting in fibers in a direct integral decomposition of K=L2(Ω,P(dω);H), (Ω,P) being the underlying probability space. Such a formulation may turn out to be useful in providing a rigorous background for theoretical physical methods employed in treating random systems. As an example a rigorous definition of the mass operator Σ(z) is presented and the formulation of transport equations in a functional analytic setting is briefly indicated. In case there is translational invariance in the probability law the operator H is shown to be unitarily equivalent to an operator Hˆ, which can be decomposed, Hˆ→Hˆ(k), in a way that diagonalizes p. Thus problems about averaged time evolution can be formulated entirely in terms of Hˆ(k) in L2(Ω,P).
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    Journal of Mathematical Physics 35 (1994), S. 146-172 
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    Notes: A non-Lagrangian approach to three-dimensional field theory is discussed. The general classification and construction of Lorentz-covariant, local fields carrying massless and massive particles of arbitrary spin are presented. The "minimal'' massive (massless) fields for (half) integer spin are explicitly constructed. Following Weinberg's analysis in the 4D case it is shown that the gauge symmetry for massive fields results from the group theoretical considerations and the condition that there should exist a long-range interaction.
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    Journal of Mathematical Physics 35 (1994), S. 1-12 
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    Notes: A stochastic dilation of master equation for a maser is constructed using the technique of quantum stochastic differential equations with unbounded operator coefficients. The associated quantum Markov flow is characterized and turns out to extend a classical birth and death process. Furthermore, the stationary distribution of the aforementioned process, which is connected to the stationary state of the micromaser, is obtained in a closed form.
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    Journal of Mathematical Physics 35 (1994), S. 237-254 
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    Notes: Using the shift vector method a large class of self-dual lattices of dimension (l,l) is obtained, which has a one to one correspondence with modular invariants of free bosonic theory compactified on a coroot lattice of a rank l Lie group. Then a large number of modular invariants of affine Lie algebras are derived explicitly. Two applications of this method are a direct derivation of the D series of SU(N) and a new proof for the A-D-E classification of the SU(2)k partition functions.
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    Journal of Mathematical Physics 35 (1994), S. 310-320 
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    Notes: Invariant connections with torsion on simple group manifolds S are studied and an explicit formula describing them is presented. This result is used for the dimensional reduction in a theory of multidimensional gravity with curvature squared terms on M4×S. The potential of scalar fields emerging from extra components of the metric and torsion is calculated, and the role of the torsion for the stability of spontaneous compactification is analyzed.
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    Journal of Mathematical Physics 35 (1994), S. 359-363 
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    Notes: The consequence of imposing stationary constraints on first-order hyperbolic differential systems is analyzed in detail and it is proved that such constraints can be compatible only with systems admitting zero-speed waves. The definition of constrained hyperbolicity is presented and examined with a view to its implications in the theory of wave propagation and in connection with second-order hyperbolic systems.
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    Journal of Mathematical Physics 35 (1994), S. 389-392 
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    Notes: An elementary proof is given for the connection between a system of continuous Hahn polynomials and symmetrizations of elements in the Heisenberg algebra [Bender, Mead, and Pinsky, Phys. Rev. Lett. 56, 2445 (1986); J. Math. Phys. 28, 509 (1987); Koornwinder, ibid. 30, 767 (1989)].
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    Journal of Mathematical Physics 35 (1994), S. 405-425 
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    Notes: G-graded contractions of the complex Lie algebra B2 and of its real forms are considered. It is found that a particular G=Z2×Z2 is a proper choice of the grading group preserved when graded contractions leading to Lie algebras of the kinematical groups of space–time are studied.
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    Journal of Mathematical Physics 35 (1994), S. 3631-3635 
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    Notes: In this article the new deformed boson algebra is proposed and the nonlinear shift operator for the algebra is obtained.
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    Journal of Mathematical Physics 35 (1994), S. 528-528 
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    Journal of Mathematical Physics 35 (1994), S. 96-112 
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    Notes: Unitarily implementable Bogoliubov transformations for charged, relativistic bosons and fermions are discussed, and explicit formulas for the two-cocycles appearing in the group product of their implementers are derived. In the fermion case this provides a simple field theoretic derivation of the well-known cocycle of the group of unitary operators on a Hilbert space modeled on the Hilbert–Schmidt class and closely related to the loop groups. In the boson case the cocycle is obtained for a similar group of pseudo-unitary (symplectic) operators. Formulas are also given for the phases of one-parameter groups of implementers and, more generally, families of implementers which are unitary propagators with parameter-dependent generators.
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    Journal of Mathematical Physics 35 (1994), S. 527-527 
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    Journal of Mathematical Physics 35 (1994), S. 2804-2817 
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    Notes: Based on the representation theory of the q-deformed Lorentz and Poincaré symmetries q-deformed relativistic wave equation are constructed. The most important cases of the Dirac, Proca, Rarita–Schwinger, and Maxwell equations are treated explicitly. The q-deformed wave operators look structurally like the undeformed ones but they consist of the generators of a noncommutative Minkowski space. The existence of the q-deformed wave equations together with previous results on the representation theory of the q-deformed Poincaré symmetry solve the q-deformed relativistic one particle problem.
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    Journal of Mathematical Physics 35 (1994), S. 2818-2827 
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    Notes: It is argued here that the global anomaly of SU(2) Weyl fermions is related to the residual topological properties of massive Dirac fermions leading to the topological index corresponding to the fermion number. As in the case of the chiral anomaly with Dirac fermions, the SU(2) anomaly with a Weyl fermion vanishes when the effect of this topological property is taken into account in the Lagrangian formulation.
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    Journal of Mathematical Physics 35 (1994), S. 2828-2832 
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    Notes: It is shown that the asymptotic state space in the BRST quantization is a Krein space. The equivalence of the representations, which are realized by the asymptotic fields in such spaces, is proved.
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    Journal of Mathematical Physics 35 (1994), S. 2833-2837 
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    Notes: The one dimensional boson–fermion model is introduced and the general eigenstates of the model Hamiltonian are constructed. The Cooper-pairlike bound state is given. It is concluded that the present model is exactly soluble via the Bethe ansatz.
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    Journal of Mathematical Physics 35 (1994), S. 2856-2877 
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    Notes: A Hamiltonian formalism is set up for nonlocal Lagrangian systems. The method is based on obtaining an equivalent singular first order Lagrangian, which is processed according to the standard Legendre transformation and then, the resulting Hamiltonian formalism is pulled back onto the phase space defined by the corresponding constraints. Finally, the standard results for local Lagrangians of any order are obtained as a particular case.
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    Journal of Mathematical Physics 35 (1994), S. 2902-2913 
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    Notes: The relation between two-dimensional integrable systems and four-dimensional self-dual Yang–Mills equations is considered. Within the twistor description and the zero-curvature representation a method is given to associate self-dual Yang–Mills connections with integrable systems of the Korteweg–de Vries and nonlinear Schrödinger type or principal chiral models. Examples of self-dual connections are constructed that as points in the moduli do not have two independent conformal symmetries.
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    Journal of Mathematical Physics 35 (1994), S. 3109-3116 
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    Notes: The heat coefficients related to the Laplace–Beltrami operator defined on the hyperbolic compact manifold H3/Γ are evaluated in the case in which the discrete group Γ contains elliptic and hyperbolic elements. It is shown that while hyperbolic elements give only exponentially vanishing corrections to the trace of the heat kernel, elliptic elements modify all coefficients of the asymptotic expansion, but the Weyl term, which remains unchanged. Some physical consequences are briefly discussed in the examples.
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    Journal of Mathematical Physics 35 (1994), S. 3190-3210 
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    Notes: The algebra Xi of nonlinear (local and nonlocal) differential operators, acting on the ring of analytic functions, is studied. It is shown in particular that this space splits into 3×2 special subalgebras ∑jr, j=0,±1, r=±1. Each subalgebra is completely specified by quantum numbers s and (p,q) describing the conformal spin, and the lowest and the highest degrees, respectively. The algebra ∑++ (and its dual ∑−−) of local (pure nonlocal) differential operators is used to calculate the general expression of the Gelfand–Dickey bracket and the Wn-symmetry Poisson, one in terms of a set of spin j canonical fields uj, 2≤j≤n and a nonlinear u-cubic dependent differential operator D(n,i,j;z,u). The explicit form of this operator is worked out. Other remarkable features are also discussed.
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    Journal of Mathematical Physics 35 (1994), S. 3211-3229 
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    Notes: The covariant differential calculus on the quantum vector space CNq leads to q deformations of the canonical commutation relations and of the Weyl algebra. If the deformation parameter q is of modulus one, integrable ("well-behaved'') representations of this twisted canonical commutation relations by self-adjoint operators on a Hilbert space are defined and studied. All faithful irreducible integrable representations of the real q-Weyl algebra are classified up to unitary equivalence.
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    Journal of Mathematical Physics 35 (1994), S. 3230-3249 
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    Notes: The solution of the two-dimensional scalar Helmholtz wave equation is discussed in a domain consisting of N+2 boundaries wherein there is one central boundary and N circles are located symmetrically within a closed space such that the system as a whole has N-fold rotational symmetry. The general solution for this complicated domain has been found to reproduce the results corresponding to simpler structures with lesser number of boundaries. Furthermore, the rotational symmetry of the system is exploited to obtain the group-theoretic solution of the composite problem leading to symmetry-induced simplifications. The numerical results pertaining to the lowest and some of the higher modes are presented for N=1 and 2. The plot of the lowest eigenvalue as a function of eccentricity exhibits maximum for symmetric as well as antisymmetric modes whenever there are at least two inner boundaries. This behavior has universal character in the sense that the maximum shows up irrespective of the values of the geometrical parameters other than the eccentricity. The computations involving the group-theoretic aspects have been found to have fast convergence highlighting the significance of our approach to deal with the solution for a class of multidisciplinary problems coming within the purview of the Helmholtz equation.
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    Journal of Mathematical Physics 35 (1994), S. 2045-2056 
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    Notes: It is shown that in the regular viz. phase space representation of a Moyal quantization real polynomials in the phase space variables become essentially self-adjoint operators, and that functions in the positive cone of monomials of even power in the phase space variables yield positive operators. Conditions on the boundedness of Moyal operators are given. The classical limit of a Moyal operator is defined and used to relate classical with quantum entropy.
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    Journal of Mathematical Physics 35 (1994), S. 2057-2063 
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    Notes: Making use of some results concerning the theory of partitions, relevant in number theory, the complete asymptotic behavior, for large N, of the level density of states for a parabosonic string is derived. It is also pointed out that there is a similarity between parabosonic strings and membranes.
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    Journal of Mathematical Physics 35 (1994), S. 2064-2073 
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    Notes: The Dirac equation for an electron on a curved space–time may be viewed as an eigenvalue problem for the Dirac operator on the spinor fields of the space–time. A general eigenvalue problem for the Dirac operator on a metric manifold M in terms of spinor and tangent fields defined via the Clifford algebra is derived herein. Then it is shown how to solve the Dirac eigenvalue problem on an imbedded submanifold N, of codimension one in M, by solving an eigenvalue equation on M. This is applied to the case of a sphere or pseudosphere imbedded in flat space. Eigenvalues and eigenfunctions for the Dirac operator on any sphere or pseudosphere are determined. In particular, when the pseudosphere is a space–time, the Dirac equation for a free lepton in this space–time can be solved. The resulting mass spectrum is discrete and depends on the curvature of the space–time.
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    Journal of Mathematical Physics 35 (1994), S. 2229-2232 
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    Notes: A nonperturbative but absolutely convergent algorithm is applied to the determination of the eigenfunctions of Hamiltonian with a singular potential of the form H=−d2/dx2+x2+λ/xα in the domain [0, ∞] which obey Dirichlet boundary conditons. The formal structure of the algorithm is identical to that of the Lanczos algorithm when it is formally extended to self-adjoint operators.
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    Journal of Mathematical Physics 35 (1994), S. 2218-2228 
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    Notes: The circular-sector quantum-billiard problem is studied. Numerical evaluation of the zeros of first-order Bessel functions finds that there is an abrupt change in the nodal-line structure of the first excited state of the system (equivalently, second eigenstate of the Laplacian) at the critical sector-angle θc=0.354π. For sector-angle θ0, in the domain 0〈θ0〈θc, the nodal curve of the first excited state is a circular-arc segment. For θc〈θ0≤π, the nodal curve of the first excited state is the bisector of the sector. Otherwise nondegenerate first excited states become twofold degenerate at the critical-angle θc. The ground- and first-excited-state energies (EG,E1) increase monotonically as θ0 decreases from its maximum value, π. A graph of E1 vs θ0 reveals an inflection point at θ0=θc, which is attributed to the change in Bessel-function contribution to the development of E1. A proof is given for the existence of a common zero for two Bessel functions whose respective orders differ by a noninteger. Application of these results is made to a number of closely allied quantum-billiard configurations.
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    Journal of Mathematical Physics 35 (1994), S. 2259-2269 
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    Notes: The symplectic structure and Hamiltonian dynamics for a class of Grassmannian manifolds are described here. Using the two-dimensional sphere (S2) and disc (D2) as illustrative cases, we write their path integral representations using coherent state techniques. These path integrals can be evaluated exactly by semiclassical methods, thus providing examples of the localization formula. Along the way, we also give a local coordinate description for a class of Grassmannians.
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    Journal of Mathematical Physics 35 (1994), S. 2233-2258 
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    Notes: In this paper, the classical two-fluid model for superconductors is used to determine the time-dependent partial differential equations (PDEs) which govern wave propagation in superconducting media. These equations are then solved exactly in both two and three dimensions using localized wave solutions rather than the traditional eigenfunction solutions. We have applied these localized wave solutions to the problem of a symmetric superconducting slab, neglecting the normally conducting current density, and found that the resultant focus wave mode magnetic field solution expels flux from the interior of the slab and can regain its initial amplitude as it travels along the surface of the slab. A comparison of the transverse part of the localized wave solution with the transverse part of the more usual plane wave solution shows remarkable agreement.
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    Journal of Mathematical Physics 35 (1994), S. 2270-2281 
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    Notes: A geometrical formalism of non-Abelian gauge theory with a topological term is constructed here with special reference to the role of Abelian gauge structures in non-Abelian theories. It is shown that when fermionic currents are written in chiral forms, we can take into account the disconnected gauge group which helps us to formulate a non-Abelian gauge structure so that the theory becomes asymptotically free.
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    Journal of Mathematical Physics 35 (1994), S. 2282-2296 
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    Notes: The eigenvalue problem for the Calogero–Sutherland–Moser model is solved exactly. The complete set of eigenfunctions is given by the homogeneous Laurent polynomials. Its subset consisting of polynomial solutions is expressed by the generalized Schur functions which correspond to the λ-deformed general linear group.
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    Journal of Mathematical Physics 35 (1994), S. 2309-2314 
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    Notes: The Dirac equation is rewritten in the framework of a Clifford algebra, revealing some new features, of which the most interesting is the U(1)×SU(2) symmetry of the corresponding Lagrangian.
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    Journal of Mathematical Physics 35 (1994), S. 2297-2308 
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    Notes: The Robertson–Schrödinger uncertainty relation for two observables A and B is shown to be minimized in the eigenstates of the operator λA+iB, λ being a complex number. Such states, called generalized intelligent states (GIS), can exhibit arbitrarily strong squeezing of A or B. The time evolution of GIS is stable for Hamiltonians which admit linear in A and B invariants. Systems of GIS for the SU(1,1) and SU(2) groups are constructed and discussed. It is shown that SU(1,1) GIS contain all the Perelomov coherent states (CS) and the Barut and Girardello CS while the spin CS are a subset of SU(2) GIS. CS for an arbitrary semisimple Lie group can be considered as a GIS for the quadratures of the Weyl generators.
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    Journal of Mathematical Physics 35 (1994), S. 2315-2335 
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    Notes: The wave-splitting Green's function approach to one-dimensional electromagnetic inverse problems of simultaneous reconstructions with different types of scattering data is analyzed. Theorems of existence, uniqueness, and stability are given for the following types of scattering data: reflection and transmission data for one round trip, two-sided reflection data for one round trip, reflection data for two round trips.
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    Journal of Mathematical Physics 35 (1994), S. 2430-2437 
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    Notes: In this work the Dirac equation for massless left-handed neutrinos has been set up for the Vaidya background which describes the exterior of a spherically symmetric radiating source. With a particular Ansatz for the form of both the spinor and the background metric, analytical solutions of the Dirac equation are obtained. The coordinates in which the particular Vaidya metric assumes diagonal form are also found.
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    Journal of Mathematical Physics 35 (1994), S. 2448-2462 
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    Notes: The complete solution of Einstein's gravitational equations with a vacuum–vacuum Kerr–Schild pencil of metrics gab+Vlalb is obtained. Our result generalizes the solution of the Kerr–Schild problem with a flat metric gab (represented by the Kerr theorem) to the case when gab is the metric of a curved space-time.
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    Journal of Mathematical Physics 35 (1994), S. 2485-2489 
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    Notes: In this comment we obtain two types of solutions of the graded Yang–Baxter equation whose symmetry group is GLqs(1||1).
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    Journal of Mathematical Physics 35 (1994), S. 2463-2484 
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    Notes: It is shown that Bäcklund transformations (BTs) and zero-curvature representations (ZCRs) of systems of partial differential equations (PDEs) are closely related. The connection is established by nonlinear representations of the symmetry group underlying the ZCR which induce gauge transformations relating different BTs. This connection is used to construct BTs from ZCRs (and vice versa). Furthermore a procedure is outlined which allows a systematic search for ZCRs of a given system of PDEs.
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    Journal of Mathematical Physics 35 (1994), S. 2516-2538 
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    Notes: Using the supersymplectic framework of Berezin, Kostant, and others, two types of supersymmetric extensions of the Schrödinger algebra (itself a conformal extension of the Galilei algebra) were constructed. An ‘I-type' extension exists in any space dimension, and for any pair of integers N+ and N−. It yields an N=N++N− superalgebra, which generalizes the N=1 supersymmetry Gauntlett et al. found for a free spin-1/2 particle, as well as the N=2 supersymmetry of the fermionic oscillator found by Beckers et al. In two space dimensions, new, ‘exotic' or ‘IJ-type' extensions arise for each pair of integers ν+ and ν−, yielding an N=2(ν++ν−) superalgebra of the type discovered recently by Leblanc et al. in nonrelativistic Chern–Simons theory. For the magnetic monopole the symmetry reduces to o(3)×osp(1/1), and for the magnetic vortex it reduces to o(2)×osp(1/2).
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    Journal of Mathematical Physics 35 (1994), S. 2539-2551 
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    Notes: For various values of the parameters in the Laguerre random matrix ensemble, the distribution of the smallest eigenvalue and the scaled n-level distribution function are calculated exactly in terms of generalized hypergeometric functions. In certain cases these functions are expressed as multidimensional integrals, from which asymptotic formulas are calculated and predictions of universal behavior verified.
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    Journal of Mathematical Physics 35 (1994), S. 2560-2569 
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    Notes: Algebras that represent a generalization of both quantum groups, quantum supergroups, and braided groups are defined. They are given by a pair of solutions of the Yang–Baxter equation that satisfy some additional conditions. Several examples are presented.
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    Journal of Mathematical Physics 35 (1994), S. 2570-2582 
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    Notes: A study of the superconformal covariantization of superdifferential operators defined on (1||1) superspace is presented. It is shown that a superdifferential operator with a particular type of constraint can be covariantized only when it is of odd order. In such a case, the action of superconformal transformation on the superdifferential operator is nothing but a Hamiltonian flow defined by the corresponding supersymmetric second Gelfand–Dickey bracket. The covariant form of a superdifferential operator of odd order is given.
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    Journal of Mathematical Physics 35 (1994), S. 1513-1521 
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    Notes: The Wigner–Poisson (WP) system (or quantum Vlasov–Poisson system) is modified to include dissipative terms in the Hamiltonian. By utilizing the equivalence of the WP system to the Schrödinger–Poisson system, global existence and uniqueness are proved and regularity properties are deduced. The proof differs somewhat from that for the nondissipative case treated previously by Brezzi–Markowich and Illner et al.; in particular the Hille–Yosida Theorem is used since the linear evolution is not unitary, and a Liapunov function is introduced to replace the energy, which is not conserved.
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    Journal of Mathematical Physics 35 (1994), S. 1532-1548 
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    Notes: The procedure of geometric quantization is developed for completely integrable dynamical systems on manifolds with exact symplectic structure. These results are applied to Neumann's nonharmonic oscillatory system on two-dimensional sphere S2.
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    Journal of Mathematical Physics 35 (1994), S. 1555-1572 
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    Notes: A quantum logic formulation of quantum measurements is introduced using Boolean powers of orthomodular σ-posets to the description of the coupled system. A comparison with the traditional Hilbert space approach to quantum measurements and a discussion of some important classes of measurements is given.
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    Journal of Mathematical Physics 35 (1994), S. 1549-1554 
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    Notes: It is shown that the polynomials introduced recently by Aldaya, Bisquert, and Navarro-Salas [Phys. Lett. A 156, 381 (1991)] in connection with a relativistic generalization of the quantum harmonic oscillator can be expressed in terms of Gegenbauer polynomials. This fact is useful in the investigation of the properties of the corresponding wave function. Some examples are given, in particular, related to the asymptotic behavior and to the distribution of zeros of the polynomials for large quantum numbers.
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    Journal of Mathematical Physics 35 (1994), S. 1573-1596 
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    Notes: Spinor fields are studied in infinite, topologically multiply connected Robertson–Walker cosmologies. Unitary spinor representations for the discrete covering groups of the spacelike slices are constructed. The spectral resolution of Dirac's equation is given in terms of horospherical elementary waves, on which the treatment of spin and energy is based in these cosmologies. The meaning of the energy and the particle–antiparticle concept is explained in the context of this varying cosmic background. Discrete symmetries, in particular inversions of the multiply connected spacelike slices, are studied. The violation of the unitarity of the parity operator, due to self-interference of P-reflected wave packets, is discussed. The violation of the CP and CPT invariance—already on the level of the free Dirac equation on this cosmological background—is pointed out.
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    Journal of Mathematical Physics 35 (1994), S. 1710-1731 
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    Notes: Integrable systems that are connected with orthogonal separation of variables in complex Riemannian spaces of constant curvature are considered herein. An isomorphism with the hyperbolic Gaudin magnet, previously pointed out by one of the authors, extends to coordinates of this type. The complete classification of these separable coordinate systems is provided by means of the corresponding L matrices for the Gaudin magnet. The limiting procedures (or ε calculus) which relate various degenerate orthogonal coordinate systems play a crucial role in the classification of all such systems.
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    Journal of Mathematical Physics 35 (1994), S. 1738-1754 
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    Notes: The properties of discrete nonlinear symmetries of integrable equations are investigated. These symmetries are shown to be canonical transformations. On the basis of the considered examples, it is concluded, that the densities of the conservation laws are changed under these transformations by spatial divergencies.
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    Journal of Mathematical Physics 35 (1994), S. 1850-1880 
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    Notes: In the present work mathematical aspects of determining the local instability parameters are focused on by using invariant characteristics of the internal Riemannian geometry with the Jacobi metric (in principle, for Hamiltonian dynamical systems with the natural Lagrangian). First, it is shown that the Ricci scalar indeed measures the sectional curvature averaged upon all two-directions. Second, necessary and sufficient criteria for non-negativity and of nonpositivity of the sectional curvature for any system with the natural Lagrangian are given. Third, analytical formulas allowing us to compute the separation rate of nearby trajectories are given. Fourth, it is shown that for any collisionless problem of n gravitationally bounded bodies, the sectional curvature in every direction is negative if n tends to infinity.
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    Journal of Mathematical Physics 35 (1994), S. 1922-1930 
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    Notes: Starting from the Cayley–Hamilton identities for n×n ordinary matrices which result from the application of the corresponding Cayley–Hamilton theorem, the generic Mandelstam identities are derived. The converse of the above statement is also proven. The method is extended to supermatrices and a definition of the generic Mandelstam identities in that case is proposed.
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    Journal of Mathematical Physics 35 (1994), S. 1984-1991 
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    Notes: The problem of formulating noncommutative differential geometry on multiparametric deformations of arbitrary quantum mechanical phase spaces involving bosonic or (even or odd-dimensional) fermionic variables is investigated. A suitable enlargement of the basic quadratic algebras enables one to solve all the consistency conditions on the calculus with the help of plausible ansatzes.
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    Journal of Mathematical Physics 35 (1994), S. 2013-2023 
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    Notes: Indecomposable modules with weight multiplicities ≤1 over the high rank Virasoro algebras are constructed and classified. A notion of the high rank super-Virasoro algebras is introduced. The classification of all indecomposable modules with weight multiplicities ≤1 over the high rank super-Virasoro algebras is obtained.
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    Journal of Mathematical Physics 35 (1994), S. 2024-2035 
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    Notes: Operators of the form Tβf(x)=∑Jj=1 cursive-phij(βx)f(σj +ρjx) are considered herein where β is a real parameter; σj and ρj are real numbers, and {cursive-phij} are holomorphic functions which satisfy certain additional conditions. Transfer operators of this form appear in statistical physics: the leading eigenvalue may be interpreted as the free energy of a lattice of interacting particles, as a function of inverse temperature β. A method for obtaining Taylor series expansions in β of the leading and second leading eigenvalues, and the leading eigenfunction are given. The method may be extended to find series expansions for other eigenvalues and eigenfunctions. The radii of convergence of these series have not been found. These results generalize the author's findings in another article [J. Math. Phys. 32, 2718 (1991)].
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    Journal of Mathematical Physics 35 (1994), S. 2036-2042 
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    Notes: Bilateral extensions of the hypergeometric formulas due to Minton (1970), Karlsson (1971), and Gasper (1981) are established through the decomposition of products of gamma functions into partial fractions. Some exceptional cases are demonstrated and a hypergeometric transformation from a bilateral series to a unilateral series is shown to contain Dougall's summation formula for 2H2 as its special case.
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    Journal of Mathematical Physics 35 (1994), S. 4429-4438 
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    Notes: The fundamental space ζ is defined as the set of entire analytic functions [test functions cursive-phi(z)], which are rapidly decreasing on the real axis. The variable z corresponds to the complex energy plane. The conjugate or dual space ζ' is the set of continuous linear functionals (distributions) on ζ. Among those distributions are the propagators, determined by the poles implied by the equations of motion and the contour of integration implied by the boundary conditions. All propagators can be represented as linear combinations of elementary (one pole) functionals. The algebra of convolution products is also determined. The Fourier transformed space ζ˜ contains test functions cursive-phi˜(x). These functions are extra-rapidly decreasing, so that the exponentially increasing solutions of higher-order equations are distributions on ζ˜.
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    Journal of Mathematical Physics 35 (1994), S. 6857-6874 
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    Notes: Highest weight representations of Uq(su(1,1)) with q=exp πi/N are investigated. The structures of the irreducible highest weight modules are discussed in detail. The Clebsch–Gordan decomposition for the tensor product of two irreducible representations is discussed. By using the results, a representation of SL(2,R)⊗Uq(su(2)) is also presented in terms of holomorphic sections which also have Uq(su(2)) index. Furthermore, ZN-graded supersymmetry is realized in terms of the representation. An explicit realization of Osp(1||2) via the highest weight representation of Uq(su(1,1)) with q2=−1 is given. © 1994 American Institute of Physics.
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    Journal of Mathematical Physics 35 (1994), S. 6802-6837 
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    Notes: An algebraic theory of integration on quantum planes and other braided spaces is introduced. In the one-dimensional case a novel picture of the Jackson q- integral as indefinite integration on the braided group of functions in one variable x is obtained. Here x is treated with braid statistics q rather than the usual bosonic or Grassmann ones. It is shown that the definite integral ∫x∞−x∞ can also be evaluated algebraically as multiples of the integral of a q-Gaussian, with x remaining as a bosonic scaling variable associated with the q-deformation. Further composing the algebraic integration with a representation then leads to ordinary numbers for the integral. Integration is also used to develop a full theory of q-Fourier transformation F The braided addition Δx=x⊗1+1⊗x and braided-antipode S is used to define a convolution product, and prove a convolution theorem. It is also proven that F2=S. The analogous results are proven on any braided group, including integration and Fourier transformation on quantum planes associated to general R matrices, including q-Euclidean and q-Minkowski spaces. © 1994 American Institute of Physics.
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    Journal of Mathematical Physics 35 (1994), S. 5600-5629 
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    Notes: A new lattice formulation of the continuum is presented, which can be applied to quantum or classical field theories in any space–time dimension. The method is illustrated with an application to relativistic nontopological solitons. This approach also provides an effective means of removing the spurious lattice fermion degenerate solutions for the Dirac equation.
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    Journal of Mathematical Physics 35 (1994), S. 5572-5581 
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    Notes: Bound-state solutions of several different Schrödinger equations are calculated rather efficiently using the methods of Lie algebra. In all cases considered, either of the algebras SO(3) or SO(2,1) provides a suitable framework, and there is no advantage to either choice. However, the choice of an appropriate realization of the generators is of greater significance, leading to a particularly simple solution whenever the Hamiltonian can be expressed as a linear function of the generators. In certain cases, the Hamiltonian can be expressed only as a bilinear function of the generators and only part of the bound-state spectrum can be calculated analytically to yield a finite set of so-called quasiexact solutions. These may be interrelated by means of suitable ladder operators and it is possible, though by no means necessary, to adopt a particular finite-dimensional representation of the underlying algebra. The present work emphasizes the role of similarity transformations, and the connection between the Lie method and other (generally variational) procedures which lead to the diagonalization of very large (in principle infinite-dimensional) matrices.
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    Journal of Mathematical Physics 35 (1994), S. 5820-5830 
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    Notes: A new approach to the construction of the rational solutions to the hierarchy of the Kadomtsev–Petviashvili equation is presented. The generalization of the "superposition formula'' is found to permit the construction of a new general solution from some partial ones. The features of the general polynomial factorization and the types of scattering in the many-body Calogero–Moser problem are investigated.
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    Journal of Mathematical Physics 35 (1994), S. 5850-5861 
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    Notes: The Kaup–Broer hierarchy can be considered as a certain reduction of the KP hierarchy. Upon introducing the τ-function the Kaup–Broer system can be formulated in trilinear form giving rise to Wronskian solutions. In this paper several odes of the Painlevé type obtained through reductions of the Kaup–Broer system are considered. Those odes take a trilinear form which allows a linearization.
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    Journal of Mathematical Physics 35 (1994), S. 5869-5884 
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    Notes: A class of (1+1)-dimensional integrable systems which are constrained from the Kadomtsev–Petviashvili (KP) hierarchy is studied. The bilinear equations of these integrable systems are shown first, and then by using the free fermion operators their rational and soliton solutions which are closely related to the τ functions of the KP hierarchy are obtained. The result implies that a large family of solutions of the (2+1)-dimensional KP hierarchy can be obtained by solving these (1+1)-dimensional integrable systems; in particular, the N-soliton solutions of the KP hierarchy can be deduced from the soliton solutions of these (1+1)-dimensional integrable systems.
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    Journal of Mathematical Physics 35 (1994), S. 5945-5968 
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    Notes: Consistent and covariant Lorentz and diffeomorphism anomalies are investigated in terms of the geometry of the universal bundle for gravity. This bundle is explicitly constructed and its geometrical structure will be studied. By means of the local index theorem for families of Bismut and Freed, the consistent gravitational anomalies are calculated. Covariant gravitational anomalies are shown to be related with secondary characteristic classes of the universal bundle and a new set of descent equations which also contains the covariant Schwinger terms is derived. The relation between consistent and covariant anomalies is studied. Finally a geometrical realization of the gravitational BRS, anti-BRS transformations is presented which enables the formulation of a kind of covariance condition for covariant gravitational anomalies.
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    Journal of Mathematical Physics 35 (1994), S. 6001-6017 
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    Notes: The spinorial methods are applied to the analysis of the wavelike exact solutions of the gauge theory of gravitation with dynamical torsion. The N-type plane-fronted metric-torsion-electromagnetic wave for the most general quadratic in the torsion and curvature Lagrangian with nonvanishing cosmological constant are presented. A series of non-plane-fronted exact solutions with nontrivial torsion are found also. The question about equivalence of equations of the gauge theory of gravitation and the general relativity in the vacuum torsionless case are clarified. The example of non-Einsteinian vacuum solution for exceptional choice of cosmological constant are presented.
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    Journal of Mathematical Physics 35 (1994), S. 6058-6075 
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    Notes: A new approach to the theory of polynomial solutions of q-difference equations is proposed. The approach is based on the representation theory of simple Lie algebras G and their q-deformations and is presented here for Uq(sl(n)). First a q-difference realization of Uq(sl(n)) in terms of n(n−1)/2 commuting variables and depending on n−1 complex representation parameters, ri, is constructed. From this realization lowest weight modules (LWM) are obtained which are studied in detail for the case n=3 (the well-known n=2 case is also recovered). All reducible LWM are found and the polynomial bases of their invariant irreducible subrepresentations are explicitly given. This also gives a classification of the quasi-exactly solvable operators in the present setting. The invariant subspaces are obtained as solutions of certain invariant q-difference equations, i.e., these are kernels of invariant q-difference operators, which are also explicitly given. Such operators were not used until now in the theory of polynomial solutions. Finally, the states in all subrepresentations are depicted graphically via the so-called Newton diagrams.
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