Volume 28, Issue 6, June 1987
Index of content:

Representations of E _{8} and other algebras
View Description Hide DescriptionA simple procedure is given for determining whether or not an arbitrary weight of a simple Lie algebra is contained in an arbitrary irreducible representation. The procedure involves two steps, determining the Weyl class of the weight and determining the classes contained in the representation. The second step applies without alteration to all simple algebras, while the first step is given here only for E _{8}. The weights of the shortest 31 Weyl classes of E _{8} are listed in a convenient, orthogonal basis.

Generalized Euler angles as intrinsic coordinates for nonlinear spinors
View Description Hide DescriptionAmong the nonlinear spinor representations of the pseudo‐orthogonal rotation groups presented in a preceding work, those ones are considered, whose carrier spaces are isomorphic with group spaces. The general element of such groups is written as a product of one‐parameter subgroups, allowing, for the SO(ν,ν) and SO(ν+1,ν) cases, to write explicitly the nonlinear spinor components in terms of independent intrinsic coordinates satisfying the Cartan’s quadratic constraints.

Decomposition of the SO*(8) enveloping algebra under U(4)⊇U(3)
View Description Hide DescriptionThe existence of a complete set of SU(3) tensor operators in the enveloping algebra of SO*(8) is demonstrated. The analysis recasts a parallel analysis by Biedenharn and Flath [Commun. Math. Phys. 9 3, 143 (1984)] concerning an SO(6,2) model for SU(3) in the isomorphic but simpler framework of an SO*(8) model.

Indecomposable modules of the Poincaré algebra in an energy‐cyclic angular momentum basis
View Description Hide DescriptionThe universal enveloping algebraU=U_{+}U_{−}H of the Poincaré algebra iso(3,1) is considered. Infinite‐dimensional induced modules U_{+}(Γ) are studied. Explicit formulas are obtained for U_{+}(Γ) in the Poincaré–Birkhoff–Witt basis. Then a change of basis is performed to an energy‐cyclic angular momentum basis. For any Γ∈C^{2}, U_{+}(Γ) turns out to be indecomposable. It can be represented as an infinite family of interacting Verma modules of the Lorentzalgebra so(3,1). Finite‐dimensional modules can be obtained as quotient modules of U_{+}(Γ) if 2Γ∈Z^{2}.

Classes of exactly solvable nonlinear evolution equations for Grassmann variables: The normal form method
View Description Hide DescriptionA systematic procedure is presented to solve analytically differential equations for Grassmann variables with the most general nonlinearity. The method consists in the reduction of the original equation to its simplest form (normal form). The classes of solvable normal forms are determined only by the structure of the linear part of the original equation and are parametrized in terms of the number of critical eigenvalues.

Lie transformations, similarity reduction, and solutions for the nonlinear Madelung fluid equations with external potential
View Description Hide DescriptionThe application of Lie‐group methods to a system of coupled nonlinear partial differential equations representing what is usually called a Madelung fluid is shown. The generating operators of the transformation group that depends on five arbitrary group constants will be constructed, and all subclasses of systems of ordinary differential equations derived by similarity reduction will be presented in tabular form. Two subclasses of physical interest are investigated in detail and the similarity solutions are compared with solutions found earlier by the application of inverse scattering transform techniques to the cubic nonlinear Schrödinger equation. Similarity solutions for the Madelung equations with linear external potential Γ(x)=−f _{0} x are presented.

Some aspects of the isogroup of the self‐dual Yang–Mills system
View Description Hide DescriptionA generalized isovector formalism is used to derive the isovectors and isogroup of the self‐dual Yang–Mills (SDYM) equation in the so‐called J formulation. In particular, the infinitesimal ‘‘hidden symmetry’’ transformation, a linear system, and a well‐known Bäcklund transformation of the SDYM equation are derived in the process. Thus symmetry and integrability aspects of the SDYM system appear in natural relationship to each other within the framework of the isovector approach.

The invariant density for a class of discrete‐time maps involving an arbitrary monotonic function operator and an integer parameter
View Description Hide DescriptionWhen x _{ t } and x _{ t+1} represent two random variables, each belonging to a real interval [0,1] and being related by a first‐order difference equationx _{ t+1}=F(x _{ t }), called a discrete‐time map, the probability density distribution connected with x _{ t } can be translated into that associated with x _{ t+1}. This yields an evolution equation by means of which one can construct an infinite sequence {w _{ t }(x)‖t∈N, x∈[0,1]} starting from an integrable functionw _{0}(x) normalized to unity on [0,1]. The question of the convergence of the sequence toward a so‐called invariant density functionw(x) as t→+∞ and the problem of finding this limit were examined by a number of authors, mostly studying isolated cases. The present paper solves the problem for a class of discrete‐time maps characterized by x _{ t+1} =f (‖sn[l sn^{−} ^{1} f ^{−} ^{1}(x _{ t })]‖), l∈{2,3,...}, whereby f is a real, continuous, monotonically increasing function mapping [0,1] onto itself and sn is the usual symbol for the sinelike Jacobian elliptic function with modulus k∈[0,1] (including the sine function). Convergence is proven under very general conditions on w _{0}(x) and an explicit formula to calculate w(x) is established. Some properties of w(x) are discussed. A necessary and sufficient condition for the symmetry of w(x) about x= 1/2 is obtained and attention is also devoted to the inverse problem, leading to a reformulation of the discrete‐time map of the type cited above which corresponds to a given invariant density. The examples of practical application considered here cover almost all special cases which were treated in the literature thus far, as well as new cases.

On unitary SU(N) ordered exponentials in a strong coupling limit
View Description Hide DescriptionA construction is given of the leading, averaged output dependence of a unitary ordered exponential in the strong coupling limit of rapidly fluctuating input. While valid for any SU(N), the method does not provide ‘‘fine structure’’ corrections to the leading output behavior. Numerical illustrations are given using a simple SU(3) example.

Existence and approximation of the solutions of some nonlinear problems
View Description Hide DescriptionNonlinear equations defined by a positive definite, elliptic operator and nonlinear functions are considered. It is proved that a unique solution exists for a wider class of problems than previously determined, which may be approximated by an iterative method. These results are shown to hold for an equation arising in some reaction‐diffusion phenomena.

On the existence of infinitely many resonances in the scattering problem based on moment conditions and entire functions
View Description Hide DescriptionIt has been well proved that there are infinitely many resonances in the scattering problem provided the potential V(t) satisfies the moment conditions ∫^{∞} _{0}‖V(t)‖t ^{ n } d t =O(n ^{(1−ε)n }), ε>0, n=0,1, ... . In particular, if V(t) has a compact support then the result of Rollnik [H. Rollnik, Z. Phys. 1 4 5, 654 (1956)] and Regge [T. Regge, Nuovo Cimento, 8, 671 (1958)] is obtained and if ‖V(t)‖∼exp(−τt ^{1+ε}), τ,ε>0, we have the results of Sartori [L. Sartori, J. Math. Phys. 4, 1408 (1963)].

Do trilinear commutation relations in quantum mechanics admit coordinate space realization in three dimensions?
View Description Hide DescriptionThe trilinear commutation relations involving coordinates and momenta introduced by Wigner [E. P. Wigner, Phys. Rev. 7 7, 711 (1950)] are generalized to three dimensions. It is shown that the only realizable coordinate space representation of the momenta implies the usual bilinear commutation relations.

Electromagnetic scattering of an arbitrary plane wave from a spherical shell with a circular aperture
View Description Hide DescriptionThe problem of the scattering of an electromagnetic plane wave with arbitrary polarization and angle of incidence from a perfectly conducting spherical shell with a circular aperture is solved with a generalized dual series approach. This canonical problem encompasses coupling to an open spherical cavity and scattering from a spherical reflector. In contrast to the closed sphere problem, the electromagneticboundary conditions couple the TE and TM modes. A pseudodecoupling of the resultant dual series equations system into dual series problems for the TE and TM modal coefficients is accomplished by introducing terms that are proportional to the associated Legendre functions P ^{−m } _{0}. The solutions of the TE and TM dual series problems require the further introduction of terms proportional to P ^{−m } _{ n }, where 0≤n<m. These functions effectively complete the standard spherical harmonic basis set when an aperture is present and guarantee the satisfaction of Meixner’s edge conditions. Having generated the modal coefficients, all desired electromagnetic quantities follow immediately. Numerical results for the currents induced on the open spherical shell and for the energy density of the field at its center are presented for the case of normal incidence.

A geometric setting for internal motions of the quantum three‐body system
View Description Hide DescriptionQuantum mechanics for internal motions of the three‐body system is set up on the basis of the complex vector bundle theory. The three‐body system is called a triatomic molecule in the Born–Oppenheimer approximation. The internal states of the molecule are described as cross sections in the complex vector bundle assigned by an eigenvalue of the square of the total angular momentum operator. This bundle is equipped with a linear connection, which is a natural consequence of a geometric interpretation of the so‐called Eckart condition. The coupling of the internal motion with the rotation is understood naturally in terms of this connection. The internal Hamiltonian operator is obtained which includes the internal motion–rotation coupling and a centrifugal potential. The complex vector bundle for the triatomic molecule proves to be a trivial bundle, though the geometric setting for the internal motion is independent of whether the bundle is trivial or not.

Quantum mechanics of a charged scalar boson with respect to an observer’s past light cone
View Description Hide DescriptionAn observer whose instantaneous ‘‘here‐now’’ has Minkowski coordinates z ^{λ}(λ=0,1,2,3) can only be aware of events within or on the past light cone with vertex at z ^{λ}. In conventional quantum mechanics his current quantum state would refer to some spacelike surface containing z ^{λ}, for example, x ^{0}=z ^{0}. This is, however, a region of space‐time about which the observer can know nothing except the single event z ^{λ}, his current here‐now. The aim of the present paper is to give a version of quantum mechanics in which the intrinsically unknowable ‘‘quantum state at the present time’’ is replaced by the ‘‘quantum state on the past light cone.’’ The theory is an extension and adaptation of Dirac’s point mechanics [Rev. Mod. Phys. 2 1, 392 (1949)].

Sufficient conditions for zero not to be an eigenvalue of the Schrödinger operator
View Description Hide DescriptionIt is proved that if H=−∇^{2}+q(x)≥0, Im q=0, ‖q(x)‖≤c(1+‖x‖)^{−a }, c=const>0, a>2, then zero is not an eigenvalue of H. An example is given of H≥0, with zero a resonance (half‐bound state) and q=q(‖x‖) compactly supported and integrable. An example of a potential q=O(r ^{−} ^{2}) is known, for which H≥0 and zero is an eigenvalue. This shows that a>2 is the optimal condition for zero not to be an eigenvalue of H≥0. If the condition H≥0 does not hold and H is an operator in L ^{2}(R^{3}), then zero can be an eigenvalue even if q∈C ^{∞} _{0}. If H is an operator in L ^{2}(R^{1}) or in L ^{2}(R^{1} _{+}), R^{1} _{+} =[0,∞), then zero cannot be an eigenvalue of H provided that a>2; here conditions H≥0 and Im q=0 can be dropped. Global estimates of the Green’s function of H from below and above are given.

The Korteweg–de Vries hierarchy of isospectral transformations: Towards a general explicit expression
View Description Hide DescriptionThe structure of the Korteweg–de Vries hierarchy of evolution equations, generating isospectral transformations, is elucidated by means of a study of its recurrence relations. For the mth member of the KdV hierarchy, which can be written in the form V _{ t }=−2A _{ m+1,x }, where the A _{ i } satisfy the recurrence relationA _{ m+1,x }=V A _{ m,x }+ 1/2 A _{ m } V _{ x }− 1/4 A _{ m,x x x }, it is shown that A _{ m } is a homogeneous polynomial in ∂^{ i } V/∂x ^{ i }. A general combinatorial formula for the coefficients of all the monomials entering A _{ m }, up to a set of constants determined by means of a recurrence relation, is derived.

Schrödinger‐like equation for relativistic particles
View Description Hide DescriptionThe Dirac equation in a spherically symmetric screened Coulomb potential is transformed to a modified Schrödinger equation of the form d ^{2} u/d r ^{2}+k ^{2}(r)u=0. This transformation is induced by expressing the Dirac function as a linear combination of the function u and its derivative d u/d r. Various properties of the transformation and of the resulting equations are studied. The close similarity between the modified Schrödinger equation and the Schrödinger equation suggests that methods applied to the Schrödinger equation to derive nonrelativistic relations can be applied to the modified Schrödinger equation to derive the analogous relativistic relations. As an example, this approach is applied to the single channel quantum defect theory to give a new derivation of its relativistic form.

A geometric approach to quantum scattering with group symmetry
View Description Hide DescriptionA geometrictheory for quantum scattering when the symmetry group is semisimple is presented. This theory is seen as a generalization of the partial wave analysis. As an application of this theory, the S‐matrix elements for scattering in the Pöschl–Teller potential with symmetry group SO(1,2), Coulomb potential with SO(1,3), and a perturbed Coulomb potential with SO(2,3) are calculated. The last example may be considered as a model for heavy‐ion scattering.

Existence and observability of spinor structure
View Description Hide DescriptionA mechanism by which space‐time topological modifications could have been controlled, in the early universe or at the Planck length, to enable onset of spinor structure is investigated. This mechanism (based on a reshuffling of topological charges and related modification of characteristic classes) could provide a gravitational analog of the Aharonov–Susskind G e d a n k e n e x p e r i m e n t proposed to detect relative rotation in the universe, spinor behavior, or to keep track of the two homotopy classes of the LorentzLie group. The space‐time topology [and in particular the trivial (nontrivial) bundle structure at conformal null infinity] provide a labeling of the asymptotic Lorentz homotopy classes which originates in the first Chern class (enclosed magnetic mass) or in the parametrization of the second homology group, and gives rise to a necessary (and sufficient) condition for the existence of spinor structure. This underlines the intertwined roles of topology and curvature. The mechanism could also be viewed as an ‘‘unwinding’’ of gravitational magnetic monopoles with one asymptotic region into electric mass (black‐hole) solutions with two asymptotic regions. In such situations a discrete PT symmetry could emerge from a continous transformation. Possible implications on the CPT theorem are mentioned.