Index of content:
Volume 28, Issue 2, February 1987

Indecomposable representations of the conformal group: A nonsingular photon‐Weyl graviton system
View Description Hide DescriptionA large class of nonelementary indecomposable representations of G=SU(2,2) is constructed and the invariant integral operators are found. An example describing a photon and a linear Weyl graviton field along with some auxiliary fields is studied. A nonsingular Lagrangian for the system is given. The pure, linear Weyl gravity with a conformal invariant gauge fixing condition arises as a particular case.

Clebsch–Gordan coefficients for SU(5) unification models
View Description Hide DescriptionThe Clebsch–Gordan coefficients for the product (1001)⊗(1001), where (1001) is the adjoint representation of SU(5), with respect to the group basis and the subgroup basis in the reduction SU(5)⊇SU(3)×SU(2)×U(1) are computed. One of the basic tools in this computation is the exhaustive use of the Verma algorithm to find bases for the weight subspaces of dimension higher than 1. It allows for the construction of bases in a systematic way by using the so‐called Verma inequalities. Only the coefficients for the dominant weights are calculated. The other ones can be obtained by using the elements of finite order (charge conjugation operators) of SU(5).

Finite‐dimensional representations of the Lie superalgebra sl(1,3) in a Gel’fand–Zetlin basis. II. Nontypical representations
View Description Hide DescriptionAll nontypical irreducible representations of the special linear Lie superalgebra sl(1,3) are considered. Explicit expressions for the transformation of the basis under the action of the generators are given. The results of this paper together with those obtained in Paper I [T. D. Palev, J. Math. Phys. 2 6, 1640 (1985)] solve the problem of the finite‐dimensional irreducible representations of sl(1,3). The results are compared with those obtained by the Young supertableau technique. A mapping of the supertableau basis on the Gel’fand–Zetlin basis is given.

Regular subalgebras of Lie superalgebras and extended Dynkin diagrams
View Description Hide DescriptionUsing the method of extended Cartan matrices and extended Dynkin diagrams, a classification of maximal regular semisimple subalgebras of the basic classical Lie superalgebras is obtained. Especially in the case of exceptional Lie superalgebras, some curious inclusion relations are discovered.

Differential‐difference AKNS equations and homogeneous Heisenberg algebras
View Description Hide DescriptionThe well‐known evolution equations associated to the homogeneous Heisenberg algebras of Kac–Moody algebrasg ^{(} ^{1} ^{)} (AKNS systems) are extended by differential‐difference equations that can be written in zero curvature form.

On invariance properties of the wave equation
View Description Hide DescriptionA complete group classification is given of both the wave equationc ^{2}(x)u _{ x x }−u _{ t t }=0 (I) and its equivalent system v _{ t }=u _{ x }, c ^{2}(x)v _{ x }=u _{ t } (II) when the wave speed c(x)≠const. Equations (I) and (II) admit either a two‐ or four‐parameter group. For the exceptional case, c(x)=(A x+B)^{2}, equation (I) admits an infinite group. Equations (I) and (II) do not always admit the same group for a given c(x): The group for (I) can have more parameters or fewer parameters than that for (II); moreover, the groups can be different with the same number of parameters. Separately for (I) and (II), all possible c(x) that admit a four‐parameter group are found explicitly. The corresponding invariant (similarity) solutions are considered. Some of these wave speeds have realistic physical properties: c(x) varies monotonically from one positive constant to another positive constant as x goes from −∞ to +∞.

On the inverse problem and prolongation structure for the modified anisotropic Heisenberg spin chain
View Description Hide DescriptionA modified form of the anisotropic Heisenberg spin chain has been considered. By use of the prolongation structure technique of Wahlquist and Estabrook, the elliptic Lax pair associated with this equation has been deduced. A variant of the Reimann–Hilbert problem on the torus is used to indicate the way to the solution of the inverse problem.

Singular points from Taylor series
View Description Hide DescriptionA simple and accurate method is developed for calculating singular points from Taylor series. It consists of finding the least‐squares deviation of the Taylor coefficients from a proposed asymptotic form. Sequences are obtained that converge quickly to the closest singularity to the origin. Some simple mathematical examples and physically interesting eigenvalue problems are discussed to illustrate the procedure. The branch points of the eigenvalues for the solutions of period 2π of the Mathieu equation and those of the Stark shifts for rigid symmetric‐top molecules, which were not obtained before, are shown.

Fractional approximation to elliptic functions
View Description Hide DescriptionThe Jacobi functions sn(x/m) have been approximated by a quotient of polynomials of first to fourth degrees. The method used here is an extension to nonlinear differential equations of one previously published for first‐ and second‐order linear differential equations. That method uses power series and asymptotic expansions simultaneously. The accuracy here obtained is very good (the absolute error is lower than 10^{−} ^{8} for m<0.8) except for values of the parameter m near 1. They are much better for several orders of magnitude than those based on the Padé method for the same number of parameters to be determined.

Quadratures for self‐dual GL(2,C) Yang–Mills fields
View Description Hide DescriptionIt is the purpose of this paper to show that the GL(2,C) Yang–Mills equations can be solved in terms of integrals over the characteristic initial data. The method is based on showing that enough gauge freedom exists in the choice of characteristic initial data so that the data can always be put into either upper or lower triangular form. With triangular form data the Sparling equation (a linear first‐order equation equivalent to the self‐dual Yang–Mills equations) can be solved by explicit quadratures.

Comment on a paper by Z‐Z. Zhong [J. Math. Phys. 2 6, 404 (1985)]
View Description Hide DescriptionThe paper of Zhong [J. Math. Phys. 2 6, 404 (1985)], though very interesting, did contain errors of both scientific and historical fact that should be corrected for the record.

Multiplicative stochastic processes involving the time derivative of a Markov process
View Description Hide DescriptionThe characteristic functional of the derivative φ̇(t) of a Markov process φ(t) and the related multiplicative process σ(t), which obeys the stochastic differential equationiσ̇(t)=( A+φ̇(t)B )σ(t), have been studied. Exact equations for the marginal characteristic functional and the marginal average of σ(t) are derived. The first equation is applied to obtain a set of equations for the marginal moments of φ̇(t) in terms of the prescribed properties of φ(t). It is illustrated by an example how these equations can be solved, and it is shown in general that φ̇(t) is delta correlated, with a smooth background. The equation of motion for the marginal average of σ(t) is solved for various cases, and it is shown how closed‐form analytical expressions for the average 〈σ(t)〉 can be obtained.

On the integrability of multidimensional nonlinear evolution equations
View Description Hide DescriptionThe integrability‐test scheme of Chen, Lee, and Liu [H. H. Chen, Y. C. Lee, and C. S. Liu, Phys. Scr. 2 0, 490 (1979)] from one‐space dimension to multispace dimensions is generalized. The temporal equation of the Lax pair is still the linearized perturbed equation that defines the symmetries. But the spectral operator in the Lax pair is no longer the linear recursion operator for symmetries. The absence of the linear recursion operator for symmetries in higher spatial dimensions therefore presents no direct obstacle to the Chen–Lee–Liu test scheme. The Kadomtsev–Petviashvili equation is shown as an example.

Continuity of scattering data for particles on the line with directed repulsive interactions
View Description Hide DescriptionThe mapping into the remote future for particles on the line interacting by repulsive directed forces is considered. Under suitable assumptions on the forces, it is proved that the mapping (which assigns the scattering data to the initial data) is continuous on a nonempty open set in the phase space.

Direct and inverse scattering in the time domain for a dissipative wave equation. III. Scattering operators in the presence of a phase velocity mismatch
View Description Hide DescriptionThe direct scattering problem for an inhomogeneous lossy medium is examined for the one‐dimensional case in which the phase velocity profile is discontinuous at the boundaries of the medium. Scattering operators (or impulse responses) and propagation operators are defined and equations that govern their behavior are developed. Knowledge of the scattering kernels for one round trip in the medium implies that the scattering kernels can be determined on any time interval. Numerical examples are presented. It is also shown that this scattering problem is reducible to one in which there are no phase velocity mismatches. This reduction provides considerable numerical advantage in the solution of the direct scattering problem. The inverse problem is examined in a companion paper.

Implementability of gauge transformations and quantization of fermions in external fields
View Description Hide DescriptionQuantization of fermions in an external soliton field, leading to a representation of the canonical anticommutation relation (CAR), which is inequivalent to the representation connected to the massive Dirac operator, is studied. Classes of gauge and axial gauge transformations that can be unitarily implemented are determined. In the latter case quantization conditions for gauge functions are obtained; integers entering can be interpreted as winding numbers.

Transition amplitude spaces
View Description Hide DescriptionMotivated by physically plausible axioms, the concept of a transition amplitude space (tas) is defined. Various connections between this framework and other axiomatic approaches to the foundations of quantum mechanics are derived. In particular, transition probability spaces, the algebraic approach, the operational statistics and quantum logic approaches, and traditional Hilbert spacequantum mechanics are considered. It is shown that a tas always admits a Hilbert space representation. Results are obtained concerning isomorphisms and automorphisms for tas’s. The concept of an A‐form is introduced and the relationship between certain A‐forms and bounded linear operators on a Hilbert space are studied. Finally, it is shown that sums and tensor products of tas’s can be formulated in a natural way.

Quantum motion on a half‐line connected to a plane
View Description Hide DescriptionIn this paper, the free motion of a particle on a manifold that consists of a one‐dimensional and a two‐dimensional part connected in one point is discussed. The class of admissible Hamiltonians is found using the theory of self‐adjoint extensions. Particular attention is paid to those Hamiltonians that allow the particle to pass through the point singularity; the reflection coefficient and other quantities characterizing scattering on the connection point are calculated. A possible application is also discussed.

The Jacobi‐matrix method in parabolic coordinates: Expansion of Coulomb functions in parabolic Sturmians
View Description Hide DescriptionPrevious analysis of the Jacobi‐matrix method based on the underlying SO(2,1) Lie algebra is extended to the Coulomb Hamiltonian in parabolic coordinates. The general solution of the generic SO(2,1) eigenvalue equation is constructed and special cases, which furnish expansions of the Coulomb functions ψ^{(±)} _{ k }(r) in a complete set of parabolic Sturmians, are discussed.

Coherent angular momentum states for the two‐dimensional oscillator
View Description Hide DescriptionCoherent angular momentum states are defined for the two‐dimensional isotropic harmonic oscillator. They share many attractive properties with the familiar (Cartesian) coherent states, but are in general distinct from those states. The probabilities of obtaining particular values for the radial and angular momentum quantum numbers follow independent Poisson distributions in the new states, but not in the old. In a quasiclassical description of the oscillator, corresponding to a given classical trajectory, the uncertainty in the angular momentum of the system is smaller if the new states are used rather than the old. The new states are the natural analogs of the coherent angular momentum states introduced for the three‐dimensional oscillator by Bracken and Leemon [A. J. Bracken and H. I. Leemon, J. Math. Phys. 2 2, 719 (1981)].