Volume 26, Issue 3, March 1985
Index of content:

Functions of infinite generalized cyclic matrices
View Description Hide DescriptionAnalytic expressions are derived for the elements or the trace of generalized cyclic matrix functions. Application to the representation of the linear difference operators with constant coefficients and periodic boundary conditions is considered, in order to express functions of such operators in closed form.

On the integrability of certain symmetric representations of the Lie algebra of SO_{0}(4,1)
View Description Hide DescriptionA proof of the existence of an essentially self‐adjoint extension of a symmetric ∼(SO_{0}(4,1)) Nelson operator, which is constructed out of the generators of a positive mass, arbitrary spin unitary irreducible representation of the Poincaré group, is presented. Our analysis of ∼(SO_{0}(4,1)) and its Lie algebra provides us with an example of an observation of Harish‐Chandra: There exist subspaces of the space of differentiable vectors of a representation of a noncompact group which are invariant under the Lie algebra, but the closures of the subspaces are not invariant under the group. The chief results of this paper should hold true for ∼(SO_{0}(n,1)). In particular, we should have a realization of an arbitrary principal series irreducible unitary representation of SO_{0}(n,1) on the direct sum of two identical unitary irreducible representation spaces of the motion group in an n‐dimensional Minkowski space, which has one timelike dimension.

A Jacobson–Morozov lemma for sp(2n, R)
View Description Hide DescriptionA variation of the lemma of Jacobson–Morozov on the imbedding of a nonzero nilpotent element of the real symplectic algebra into the split simple three‐dimensional Lie algebra is proved. The proof is algorithmic and relies on our earlier work on the theory of normal forms for the real symplectic algebra.

Representations of Kac–Moody algebras by step algebras
View Description Hide DescriptionRepresentations of a Kac–Moody algebraĝ associated to a semisimple Lie algebrag, which decompose to a direct sum of finite‐dimensional representations of the subalgebra g, are constructed using a step algebra method. The cases g=su(2), su(3) are considered in detail.

Spectral concentration for the Laplace operator in the exterior of a resonator
View Description Hide DescriptionWe consider the Laplace operator defined in the exterior of a resonator and we provide explicit estimates for the spectral concentration, in the sense of Kato, in terms of the width of the channel connecting the cavity with the exterior.

Consistent superspace integration
View Description Hide DescriptionThe standard Berezin method for integration over odd variables is combined in a new way with De Witt’s contour method for integration over even Grassmann variables to give a new method of superspace integration. It is shown that this integral, unlike the standard superspace integral, is invariant under coordinate transformations in superspace. The relation between the new method and the standard method is discussed.

Application of linked Bäcklund transformations to nonlinear boundary value problems
View Description Hide DescriptionA class of nonlinear boundary value problems is reduced to linear canonical form by a combination of Bäcklund transformations.

Some properties of hyperspherical harmonics
View Description Hide DescriptionA general formula is given for the canonical decomposition of a homogeneous polynomial of order λ in m variables into a sum of harmonic polynomials. This formula, which involves successive applications of the generalized Laplace operator, is proved in the Appendix. It is shown that the group‐theoretical method for constructing irreducible Cartesian tensors follows from the general formula for canonical decomposition. The relationship between harmonic polynomials and hyperspherical harmonics is discussed, and an addition theorem for hyperspherical harmonics is derived. An expansion of a many‐dimensional plane wave in terms of Gegenbauer polynomials and Bessel functions is derived and used to construct bicenter expansions of arbitrary functions in many‐dimensional spaces. Finally, a formula is derived for the 3λ coefficients of hyperspherical harmonics. These coefficients give the values of integrals involving the products of three harmonics.

On the hyperbolic complex linear symmetry groups and their local gauge transformation actions
View Description Hide DescriptionThe hyperbolic complex linear groups and the isomorphic relation between these groups and real linear groups are discussed. A local hyperbolic complex gauge symmetry of the hyperbolic complex sesquilinear field is equivalent to some local real gauge symmetry of the real bilinear field.

Infinitesimal null isotropy and Robertson–Walker metrics
View Description Hide DescriptionThe concept of infinitesimal null isotropy is defined for a Lorentz manifold, in terms of null sectional curvature (as defined by Harris). It is shown that infinitesimal null isotropy is equivalent to infinitesimal spatial isotropy (as defined by Karcher), and that a null‐isotropic space for which null sectional curvature is infinitesimally spatially constant must have a Robertson–Walker metric.

High‐accuracy approximation techniques for analytic functions
View Description Hide DescriptionA generalization of the familiar mesh point technique for numerical approximation of functions is presented. High accuracy and very rapid convergence may be obtained by thoughtful choice of the reference function chosen for interpolation between the mesh points. In particular, derivative operators are represented by highly nonlocal matrices; but this is no drawback when one has computing machines to perform the algebraic manipulations. Some examples are given from familiar quantum mechanical problems.

A new way for solving Laplace’s problem (the predictor jump method)
View Description Hide DescriptionThis paper presents a new method, that we call ‘‘the predictor jump,’’ for driving to a faster solution of Laplace’s equation. Some results obtained by applying this technique are compared with those that have been obtained by the traditional methods.

Construction of the second constant of motion for two‐dimensional classical systems
View Description Hide DescriptionA general method for the construction of the second constant of motion of third and fourth orders is given for two‐dimensional systems in terms of z=q _{1}+i q _{2}, and z̄=q _{1}−i q _{2}. Correspondingly, the third‐ and fourth‐order potential equations are obtained whose solutions directly provide the integrable systems. Using the Holt ansatz, the potential equation corresponding to the third‐order invariants has been reduced to a pair of potential equations whose solutions yield a large class of integrable systems.

The anti‐self‐dual Coulomb field in Minkowski space‐time
View Description Hide DescriptionThe twistor encoding of the anti‐self‐dual Coulomb field is given in terms of the space‐time connections pulled back to ^{+} and to ^{−}. This description differs considerably from that of the twistor encoding of transverse or radiation fields, which have been the only fields studied in this fashion to date. A twisted structure results on ^{+} and a topologically incomputable one on ^{−} and these are identified modulo the null lines intersecting the source world‐line.

A twistor encoding of Lienard–Wiechert fields in Minkowski space‐time
View Description Hide DescriptionThe twistor encoding of the anti‐self‐dual Lienard–Wiechert field on Minkowski space‐time yields a considerably richer structure than that of the Coulomb field encoding due to the presence of a nonzero radiation field. The combination of advanced and retarded transverse fields together with the longitudinal field and the individual aspects of these fields provides this structure. Higher‐order longitudinal moments can be incorporated so that general longitudinal fields can be given a twistor description.

Nontrivial zeros of the Wigner (3‐j) and Racah (6‐j) coefficients. I. Linear solutions
View Description Hide DescriptionSome formuals for nontrivial zeros in the 3‐j and 6‐j symbols have been found.

Three‐dimensional inverse scattering: High‐frequency analysis of Newton’s Marchenko equation
View Description Hide DescriptionWe obtain a high‐frequency asymptotic expansion of Newton’s Marchenko equation for three‐dimensional inverse scattering. We find that the inhomogeneous term contains the same high‐frequency information as does the Born approximation. We show that recovery of the potential via Newton’s Marchenko equation plus the ‘‘miracle’’ depends on low‐frequency information.

Scattering theory for extended elementary particles in nonrelativistic stochastic quantum mechanics
View Description Hide DescriptionScattering theory for extended elementary particles in stochastic phase space is studied. It is shown that the interacting Hamiltonian is equivalent to an effective potential in configuration representation. Asymptotic completeness can be studied by investigating the behavior of the effective potential. The sharp‐point limit of the extension of these particles is studied. It is also shown that scattering theory can also be studied directly in stochastic phase space in the optimal case.

Schrödinger semigroups for vector fields
View Description Hide DescriptionSuppose H is the Hamiltonian that generates time evolution in an N‐body, spin‐dependent, nonrelativistic quantum system. If r is the total number of independent spin components and the particles move in three dimensions, then the Hamiltonian H is an r×r matrix operator given by the sum of the negative Laplacian −Δ_{ x } on the (d=3N)‐dimensional Euclidean space R^{ d } plus a Hermitian local matrix potential W(x). Uniform higher‐order asymptotic expansions are derived for the time‐evolution kernel, the heat kernel, and the resolvent kernel. These expansions are, respectively, for short times, high temperatures, and high energies. Explicit formulas for the matrix‐valued coefficient functions entering the asymptotic expansions are determined. All the asymptotic expansions are accompanied by bounds for their respective error terms. These results are obtained for the class of potentials defined as the Fourier image of bounded complex‐valued matrix measures. This class is suitable for the N‐body problem since interactions of this type do not necessarily decrease as ‖x‖→∞. Furthermore, this Fourier image class also contains periodic, almost periodic, and continuous random potentials. The method employed is based upon a constructive series representation of the kernels that define the analytic semigroup {e ^{−z H }‖Re z>0}. The asymptotic expansions obtained are valid for all finite coordinate space dimensions d and all finite vector space dimensions r, and are uniform in R^{ d }×R^{ d }. The order of expansion is solely a function of the smoothness properties of the local potential W(x).

The gravitational field of a charged, magnetized, accelerating, and rotating mass
View Description Hide DescriptionThe explicit expression of a Petrov type G solution to the Einstein–Maxwell equations is given. This new solution is endowed with eight arbitrary parameters; mass, Newman–Unti–Tamburino (NUT) parameter, angular momentum, acceleration, electric and magnetic charges, and electric and magnetic field parameters.