Volume 23, Issue 3, March 1982
Index of content:

Mickelsson lowering operators for the symplectic group
View Description Hide DescriptionElementary lowering operators for the symplectic group, for which a graphical algorithm was given by Mickelsson, are obtained in the form of tensor operators. The resultant simple analytic expressions are analogous to the corresponding ones found previously for the unitary and orthogonal groups.

Boson–fermion representations of Lie superalgebras: The example of osp(1,2)
View Description Hide DescriptionA method for constructing infinite‐dimensional representations of Lie superalgebras employing boson representations of their Lie subalgebras is outlined. As an example the osp(1,2) superalgebra is considered; explicit formulae for its generators in terms of one pair of boson operators, at most one pair of fermion ones, and at most one parameter are obtained, the Casimir operator being represented by a multiple of unity. The restriction of these representations to the real form of osp(1,2) is skew‐symmetric in the even part and can be regarded as a natural generalization of skew‐symmetric representations of real Lie algebras. Some other aspects of the presented construction are discussed.

Lie‐algebraic properties of infinite‐dimensional wave equations
View Description Hide DescriptionTo an infinite‐dimensional Lorentz‐invariant wave equation of the form (α^{μ}∂_{μ}+iκ)ψ(x) = 0 is associated a Lie algebraS over C which contains so(4,C) and α^{μ}. We show by considering a certain class of equations, that in general S is an infinite‐dimensional Lie algebra. It has a structure which is quite different from that of known types of infinite‐dimensional algebras.

Remarks on certain dual series equations involving the Konhauser biorthogonal polynomials
View Description Hide DescriptionIt is observed, in the present note, that the literature contains erroneous results concerning the solutions of certain dual (and triple) equations involving series of the Konhauser biorthogonal polynomials. For example, the main results proved recently by K. R. Patil and N. K Thakare [J. Math. Phys. 18, 1724 (1977)] are shown to be invalid except in their already‐known special cases. The errors are traced to the misuse of a certain Weyl fractional integral which holds true only in the case of the classical Laguerre polynomials.

The bi‐Hamiltonian structure of some nonlinear fifth‐ and seventh‐order differential equations and recursion formulas for their symmetries and conserved covariants
View Description Hide DescriptionUsing a bi‐Hamiltonian formulation we give explicit formulas for the conserved quantities and infinitesimal generators of symmetries for some nonlinear fifth‐ and seventh‐order nonlinear partial differential equations; among them, the Caudrey–Dodd–Gibbon–Sawada–Kotera equation and the Kupershmidt equation. We show that the Lie algebras of the symmetry groups of these equations are of a very special form: Among the C ^{∞}vector fields they are generated from two given commuting vector fields by a recursive application of a single operator. Furthermore, for some higher order equations, those multisoliton solutions, which for ‖t‖→∞ asymptotically decompose into traveling wave solutions, are characterized as eigenvector decompositions of certain operators.

Singular symmetries of integrable curves and surfaces
View Description Hide DescriptionIf w = w(x,t) is a solution of w _{ t } = 6w w _{ x }−w _{ x x x }+6ε^{2} w ^{2} w _{ x } then ? = −w−ε^{−2} is also a solution. In general, integrable families and their members admit discrete symmetries whereas original systems may not.

Recurrence relations for the coefficients of perturbation expansions
View Description Hide DescriptionExact recurrence relations are derived for the coefficients of the perturbation expansion of the Schrödinger wavefunction for large classes of potentials. The terms of the eigenvalue expansion can then be expressed in terms of these coefficients which therefore allow other investigations such as the large‐order behavior of the expansion.

An identity in Riemann–Cartan geometry
View Description Hide DescriptionWe derive a new Gauss–Bonnet type identity in Riemann‐Cartan geometry: (−g)^{1/2}ε^{μνλρ} (R _{μνλρ} + (1/2) C ^{α} _{μν} C _{αλν}) = ∂_{μ} (−(−g)^{1/2}ε^{μνλρ} C _{μνλρ}), where C ^{α} _{μν} is the torsion tensor.

Theta functions, Gaussian series, and spatially periodic solutions of the Korteweg–de Vries equation
View Description Hide DescriptionIt has been shown by Novikov [Funct. Anal. Appl. 8, 236 (1974)], Dubrovin e t a l. [Russian Math. Surveys 31, 59 (1976)], Lax [Commun. Pure Appl. Math. 28, 141 (1975)], McKean and van Moerbeke [Inv. Math. 30, 217 (1975)], and others that the nonlinear evolution equations which admit solitary waves also have spatially periodic exact solutions (’’polycnoidal waves’’) which can be expressed in terms of multidimensional Riemann theta functions. Here, it is shown that via Poisson summation, the Fourier series that define the theta functions can be transformed into an infinite series of Gaussian functions. Because the lowest terms of the Gaussian series generate the usual solitary waves, it is possible to intimately explore the relationship between solitary waves and these spatially periodic ’’polycnoidal’’ waves. Also, by using the Gaussian series, one can perturbatively calculate phase velocities and wave structure for the ’’polycnoidal’’ wave even in the strongly nonlinear regime for which the soliton (or multisoliton) is the lowest order approximation. It is further shown that the Fourier series and the complementary Gaussian series both converge so rapidly in the intermediate regime of moderate nonlinearity that one may loosely state that a solitary wave is almost a linear wave, and a linear wave almost a soliton. Thus, by using both series together, one can obtain a very complete description of these stable, finite amplitude, periodic solutions. For expository simplicity, this first discussion of the Gaussian series approach to ’’polycnoidal’’ waves will concentrate on the most elementary example: the ordinary ’’cnoidal’’ wave of the Korteweg–de Vries equation. The great virtue of the Poisson method, however, is that it extends almost trivially to other equations (the Nonlinear Schrödinger equation, the Sine–Gordon equation, and a multitude of others) and also to periodic solutions of these equations that are describable in terms of higher dimensional theta functions (’’polycnoidal’’ waves). The next to last section proves a number of generalizations of the theorems of Hirota [Prog. Theor. Phys. 52, 1498 (1974)], applicable both to ’’cnoidal’’ and ’’polycnoidal’’ solutions without restriction, and explains how these extensions will work.

String mechanics based on 2‐forms
View Description Hide DescriptionNambu invented a mechanics for strings by replacing the fundamental 1‐form p _{ i } d q ^{ i }−H d t of Hamiltonian mechanics by a certain 2‐form. We study the mechanics corresponding to a more general 2‐form applicable to weighted strings. Our equations of motion are fully deterministic, unlike those of Nambu, which need a supplementary condition. We set up a Hamilton–Jacobi formalism closely paralleling ordinary mechanics.

Sine‐Gordon and modified Korteweg–de Vries charges
View Description Hide DescriptionThe sine‐Gordon and the integrated modified Korteweg–de Vries equations are shown to conserve the same infinite set of charges. The charges are determined by a recursion relation. As a consequence, the solutions of all the equations generated by the charges have in common all time‐independent properties.

Inverse problems for nonabsorbing media with discontinuous material properties
View Description Hide DescriptionOne‐dimensional electromagnetic and elasticinverse problems are formulated for media with discontinuous material properties. In addition, an impedence mismatch between source and medium is allowed. The measured data for either problem is shown to generate a reflection kernel which is used in the solution of the inverse problem. The solution algorithm itself is a time domain technique which is a special case of previously obtained results for absorbing media.

Feynman path integral and Poisson processes with piecewise classical paths
View Description Hide DescriptionWe prove the existence of a Feynman integral formula for gentle perturbations of the harmonic oscillator. This result is extended to Bose relativistic theory.

Theoretical basis for Coulomb matrix elements in the oscillator representation
View Description Hide DescriptionHydrogenic wave functions in the spherical and parabolic bases are shown to correspond, respectively, to a restricted set of wave functions of a four‐dimensional harmonic oscillator and its coupled pair of two‐dimensional oscillators. This correspondence provides the theoretical basis for algebraic calculations of Coulomb matrix elements in the oscillator representation.

Simple multiple explode–decay mode solutions of a two‐dimensional nonlinear Schrödinger equation
View Description Hide DescriptionMultiple similarity type explode‐decay mode solutions of a two‐dimensional (≡2D) nonlinear Schrödinger (≡NLS) equation have been obtained by the bilinear method. From the three examples of the 2D‐KdV equation, ordinary cubic 2D‐NLS equation, and the present 2D‐NLS equation, the expectation is presented such that ’’any 2D nonlinear evolution equation which has multiple soliton solutions simultaneously has simple self‐similar‐type multiple explode‐decay mode solutions so far as the equation has self‐similar symmetry.’’

A combinatoric result related to the N‐body problem
View Description Hide DescriptionIn the set of complete chains of partitions of N objects, let chains that are related through a permutation of the objects be termed equivalent. The number of equivalence classes μ_{ N } is shown to equal the Euler number ‖E _{ N−1}‖ if N is odd, and 2^{ N }(2^{ N }−1)‖B _{ N }‖/N, where B _{ N } is a Bernoulli number, if N is even. The number of elements in each class is also found. In the Yakubovskii‐type formulation of the N‐body problem in quantum mechanics, μ_{ N } is the basic number of coupled equations when all particles are identical.

A generalization of the Dirac equation to accelerating reference frames
View Description Hide DescriptionUsing a recently developed global isometry method for treating accelerating observers, the induced tangent space transformation on flat Lorentzian R ^{4} is mapped homomorphically onto a time‐dependent D ^{(1/2,0)} ⊕ D ^{(0,1/2)} representation of SL (2,C). The Dirac equation is shown to take on pseudoterms via this mapping. Eliminating the pseudoterms by identifying an affine connection, an exact analytic expression for the covariant derivative is found for general cases of arbitrary C ^{2} timelike observers. The transformation properties of the connection are shown to satisfy the conditions imposed by a general tetrad formalism. The specific case of the rotating observer is considered wherein the exact expression for the boosted Dirac equation is found.

The stationary coordinate systems in flat spacetime
View Description Hide DescriptionThe stationary metrics in flat spacetime are derived using a recent classification of the timelike Killing vector field trajectories. The metrics fall into six classes. A ’’simple’’ coordinate system from each class is selected as representative. Three of these systems are rectangular Minkowski coordinates, pseudocylindrical (’’accelerating’’) coordinates, and rotating coordinates. The remaining three appear to be new coordinate types which will be useful in exploring coordinate‐dependent effects in quantum field theory.

Singular boundaries of space–times
View Description Hide DescriptionWe give an example of a causally well‐behaved, singular space–time for which all singular‐boundary constructions which fall in a certain wide class—a class which includes both the g‐boundary and b‐boundary—yield pathological topological properties. Specifically, for such a construction as applied to this example, a singular boundary point fails to be T _{1}‐related to an event of the original space–time. This example suggests that there may not exist any useful, generally applicable notion of the singular boundary of a space–time.

The Riemann tensor, the metric tensor, and curvature collineations in general relativity
View Description Hide DescriptionThe equation x _{μν} R ^{μ} _{ λαβ}+x _{μλ} R ^{μ} _{ ναβ} = 0, where x _{μν} and R ^{μ} _{ ναβ} are the components of an arbitrary symmetric tensor and of the Riemann tensor formed from the metric tensorg _{μν}, is trivially satisfied by x _{μν} = φg _{μν}. Nontrivial solutions are important in various areas of general relativity such as in the study of curvature collineations, and also in the study of algebraic methods given by Hlavatý and Ihrig for the determination of g _{μν}, from a given set of R ^{μ} _{ ναβ}. We have found all R ^{μ} _{ ναβ} for which there exist nontrivial solutions of the above equation, and we have given the form of the x _{μν} in each case. Various examples of space–times for explicit nontrivial solutions are discussed.