Volume 33, Issue 8, August 1992
Index of content:

Realization of a class of affine Lie superalgebras
View Description Hide DescriptionIn this paper, a class of affine Lie superalgebras is presented, whose Dynkin diagrams are constructed by, as usual, extending a Dynkin diagram D of Lie algebra (or superalgebra) by adding a node ψ that corresponds to the lowest root ψ_{0} of D. However, this added node ψ does not have the same degree as ψ_{0} in Z _{2} gradation (Z _{2}=Z/2Z), i.e., ψ corresponds to an odd or even root as ψ_{0} corresponds to even or odd root. As far as we know, these Lie superalgebras did not appear in literature, and therefore we believe these are a new class of affine Lie superalgebras.

Wigner analysis and Casimir operators of SA(4,R)
View Description Hide DescriptionIn theories involving gravity, including QCD‐generated gravitylike effects in hadrons, SA(4,R) plays a role. Its single Casimir invariant and that of its SA(2,R) and SA(3,R) subgroups are evaluated. The group orbits are studied and the unitary irreducible representations are classified.

Irreducible representations of braid groups
View Description Hide DescriptionThe irreducible representations of the braid groups, described by the Young patterns, are obtained in terms of a set of commutant operators. The relations between our representations and the monodromy representations based on the minimal representation of the quantum enveloping algebraU _{ q }sl(2) are discussed briefly.

Weight diagram construction of Lax operators
View Description Hide DescriptionIt is demonstrated and proven that cyclic weight diagrams corresponding to representations of affine Lie algebras allow one to construct the associated Lax operator. The resultant Lax operator is in the Miura‐like form and generates the modified KdV equations. The algorithm is extended to the supersymmetric case.

Symmetries of the renormalization group equations
View Description Hide DescriptionThe invariance of the renormalization group equation under the action of the prolonged Lie field is examined and it is shown that the system exhibits infinite dimensional symmetry. From the characteristics, detailed solutions of the renormalization group equation is derived. As an illustration, the method is applied to QED and QCD.

One‐dimensional nonautonomous dynamical systems with exact transcendental invariants
View Description Hide DescriptionNew classes of dynamical systems for nonautonomous particle motion in one dimension are exactly solved in terms of arbitrary functions of both space and time. Besides extending considerably the class of mechanical systems with known exact solutions, the explicit dependence on space represents an improvement on previous results, such as those obtained by Lewis and Leach. This might be helpful for extending the Lewis and Symon technique of constructing the BGKY type of solutions for the Vlasov–Poisson equations.

Grassmannian Legendre functions
View Description Hide DescriptionThe Grassmannian analogues of the Legendre polonomials are obtained from the corresponding Grassmannian Hermite multinomials. The generating function, recurrence relations, and differential equation are given. In contrast to the Hermitian case there is no Berezin weight function that orthogonalizes these functions. Generalizations to the Tchebyscheff and Gegenbauer cases are also possible.

Stokes constants for a certain class of second‐order ordinary differential equations
View Description Hide DescriptionBy generalizing the coupled wave integral equations method devised by Hinton, the Stokes phenomena of the standard second‐order ordinary differential equations with the following coefficient functions q(z) are analyzed: (i) q(z)=a _{2N } z ^{2N }+∑_{ j }=−∞^{ N−1} a jz ^{ j }; (ii) q(z)=a _{2N−1} z ^{2N−1}+∑_{ j }= −∞^{ N−2} a _{ jz } ^{ j }; (iii) q(z)=∑_{ j }=0^{4} a _{ jz } ^{ j }, with a _{3}=0; and (iv) q(z)=∑_{ j }=0^{3} a _{ jz } ^{ j }, with a _{2}=0. First the relations are derived among the Stokes constants, which enable the expression of all Stokes constants in terms of only one. Furthermore, this one Stokes constant is shown to be expressed in the analytical form of a convergent infinite series as a function of the coefficients of q(z). This was done for all four cases listed above.

Partially invariant solutions of nonlinear Klein–Gordon and Laplace equations
View Description Hide DescriptionThe concept of partially invariant solutions is discussed in the framework of the group analysis of systems of partial differential equations. Using a general and systematic approach based on subgroup classification methods, nontrivial partially invariant solutions for specific equations are explicitly obtained, belonging to the NLKGE and NLLE classes in two dimensions. It is proven that the obtained partially invariant solutions are distinct from the invariant ones and from solutions obtained by other methods of reducing PDEs to ODEs.

Investigation of Painlevé property under time singularities transformations
View Description Hide DescriptionIn this paper, the relation between the Painlevé property for ordinary differential equations and some coordinate transformations in the complex plane is investigated. Two sets of transformations are introduced for which the transformations of singularities structure in the complex plane are explicitly computed. The first set acts only on the dependent variables while the second one acts also on the independent variables. It allows for defining an extended Painlevé test which is coordinates invariant. Our results are applied on various problems arising in the singularity analysis such as the problem of negative resonances and the weak Painlevé conjecture. The method for finding new first integrals for dynamical systems is also shown.

On the bi‐Hamiltonian structures of the sKdV
View Description Hide DescriptionThe first and the second Hamiltonian structures of the supersymmetric KdV equation are derived, following the conventional approach of Gel’fand and Dickey. In particular, it is shown that the nonlocal nature of the first bracket is a direct consequence of the constrained nature of the Lax operator.

The Courant–Hilbert solutions of the wave equation
View Description Hide DescriptionThe Bateman transformations which generate many new solutions from a particular solution of the scalar wave equation in a homogeneous medium are discussed. The Bateman transformations are then used to classify the Courant–Hilbert solutions of the wave equation. The latter are the product of an arbitrary function F of the phase u, a solution of the characteristic equation, and of a specific attenuation factor f that does not depend on the particular form of F. It is shown that for a given phase u, f is not unique. Consequently, there exist spherical waves F(r±ct) with an attenuation factor different from r ^{−1}.

Dynamics of solitons in a damped sine‐Hilbert equation
View Description Hide DescriptionA damped sine‐Hilbert (sH) equation is proposed. It can be linearized by a dependent variable transformation which enables one to solve an initial value problem of the equation. The N‐soliton solution is obtained explicitly and its properties are investigated in comparison with those of the N‐soliton solution of the sH equation. In particular the interaction of the two solitons is explored in detail with the aid of the pole representation. It is found that the interaction process is classified into the two types according to the initial amplitudes and positions of both solitons. In the general N‐soliton case the long‐time behavior of the solution is shown to be characterized by the positive N zeros of the Hermite polynomial of degree 2N. Finally, a linearized version of the damped sH equation is briefly discussed.

Dilation currents
View Description Hide DescriptionThe conserved currents related to conformal Killing vectors are analyzed. By considering the dilation generator of Minkowski space‐time, with the assumption that fields are static, arrived at is what is often known as the relativistic virial theorem. The application of this result to force‐free electromagnetic fields is discussed, and the result is shown how it can rule out particlelike solutions to certain field theories.

A q‐analog of the quantum central limit theorem for SU_{ q }(2)
View Description Hide DescriptionA q‐analog of the central limit theorem for SU_{ q }(2), q≳0, is studied. It is shown that the limits of the moments and q‐exponential generating functions for coherent states give for q≳1 (0<q<1) the harmonic q‐oscillator (q ^{−1}‐oscillator) introduced by Biedenharn and Macfarlane.

Energy levels of a three‐dimensional anharmonic oscillator with sextic perturbation
View Description Hide DescriptionRenormalized series version of inner product and renormalized series techniques with Hill determinant approach are used to calculate the energy eigenvalues for a three‐dimensional oscillator for several sets of parameters. Our techniques were modified to treat higher power of perturbation for two eigenstates E _{0,0,1} and E _{1,1,0} with odd parity.

On the elementary Schrödinger bound states and their multiplets
View Description Hide DescriptionThe problem of the existence of elementary bound states is discussed. A−trivial−observation that every elementary wave function ψ^{[i]}(r) is an exact bound state for an appropriate potential, V(r)=V ^{[i]}[ψ(r),r], is shown to lead to a very transparent form of the ‘‘quasiexact’’ (QE) solvability condition V ^{[i]}=V ^{[j]} for doublets and multiplets of the ψ’s. In this sense, the particular class of elementary ansätze, ψ^{[i]}(r)=r ^{λ}polynomial(r ^{2}) ×exp[r ^{μ}polynomial(r ^{2})], also defines the particular class of QE‐solvable potentials. They have an elementary nonpolynomial (rational) form, possibly also with a strongly singular−repulsive−core at the origin. The properties of these forces are discussed in detail.

Two‐body resonance scattering and annihilation of composite charged particles
View Description Hide DescriptionA model of two‐body scattering including interaction between the external (Coulomb) and internal (e.g., quark) channels is constructed and investigated. A mathematically strict description is based on extensions theory for symmetric operators. Extra (internal) channel simulates complicated structure of charged particles and generates energy‐dependent effective interaction in the external channel. The main effects of this short‐range energy‐dependent interaction in the system of charged particles (Zel’dovich effect, appearance of resonances, relative shift formula, and so on) are studied. Both for models of zero–range and nonzero–range energy‐dependent interaction stationary scattering theory is constructed. In the frames of the same method a model of a system with extra (annihilation) scattering channel is considered.

Ferromagneticity of simplicial fields on two‐dimensional compact manifolds
View Description Hide DescriptionSmooth triangulations of a compact smooth connected two‐dimensional Riemannian manifoldM are considered. The q‐simplicial fields are defined with values in the space of q‐cochains and a natural Gaussian measure is defined giving their distribution, with covariance defined essentially in terms of the combinatorial Laplacian Δ^{ c }. In the continuum limit this measure for q=0 is the free quantum fieldmeasure over M. In this case it is shown that for a certain collection of triangulations there exists a sequence of subdivisions of each triangulation such that the corresponding measure is ferromagnetic. It is also shown that for sufficiently fine subdivisions −Δ+m ^{2} I, m≳0 has nonpositive off‐diagonal elements. The proofs are obtained by a result on triangulations by simplexes with acute angles. It is also proven that the probability measures describing quantum fields on M with polynomial, trigonometric, or exponential interactions satisfy FKG inequalities.

Noncommutative geometry, chiral anomaly in the quantum projective [sl(2,C)‐invariant] field theory and jl(2,C)‐invariance
View Description Hide DescriptionThe quantum projective [sl(2,C)‐invariant] field theories with chiral anomaly [with noncommuting holomorphic and antiholomorphic sl(2,C)‐symmetries] are considered. The underlying noncommutative geometry is described. It is shown that such theories are invariant with respect to the BL (Borel–Lie) algebra jl(2,C), the central extension of the double sl(2,C)+sl(2,C). The jl(2,C)‐invariant quantum projective field theories are classified: it is proved that the jl(2,C)‐primary fields are characterized by their weights (λ,μ̄), the corresponding fusion rules are trivial [V _{λ,μ̄}][V _{λ} ^{’},μ̄^{’}]= [V _{λ+λ} ^{’},μ̄+μ̄^{’}] (though the primary fields are not mutually local) and so the category of the jl(2,C)‐invariant quantum projective field theories’ (QPFT)‐operator algebras is isomorphic to the category of R ^{2}‐graded associative algebras. The explicit form of the jl(2,C)‐primary fields is found.