JMP Nameplate
  • Volume/Page
  • Keyword
  • DOI
  • Citation
  • Advanced
   
 
 
 

Flickr Twitter iResearch App Facebook

Search Issue | RSS Feeds RSS
Previous Issue Next Issue

Mar 1982

Volume 23, Issue 3, pp. 347-471


Mickelsson lowering operators for the symplectic group

Adam M. Bincer

J. Math. Phys. 23, 347 (1982); http://dx.doi.org/10.1063/1.525372 (3 pages) | Cited 6 times

Full Text: | Download PDF

Show Abstract
Elementary 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.
Show PACS
02.20.Qs General properties, structure, and representation of Lie groups

Boson–fermion representations of Lie superalgebras: The example of osp(1,2)

Jiří Blank, Miloslav Havlíček, Pavel Exner, and Wolfgang Lassner

J. Math. Phys. 23, 350 (1982); http://dx.doi.org/10.1063/1.525373 (4 pages) | Cited 7 times

Full Text: | Download PDF

Show Abstract
A 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.
Show PACS
02.20.Qs General properties, structure, and representation of Lie groups
11.30.Pb Supersymmetry
02.20.Sv Lie algebras of Lie groups
02.20.Tw Infinite-dimensional Lie groups

Lie‐algebraic properties of infinite‐dimensional wave equations

A. Cant

J. Math. Phys. 23, 354 (1982); http://dx.doi.org/10.1063/1.525374 (3 pages) | Cited 1 time

Full Text: | Download PDF

Show Abstract
To an infinite‐dimensional Lorentz‐invariant wave equation of the form (αμ∂μ+iκ)ψ(x) = 0 is associated a Lie algebra S over C which contains so(4,C) and αμ. 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.
Show PACS
02.20.Tw Infinite-dimensional Lie groups

Remarks on certain dual series equations involving the Konhauser biorthogonal polynomials

H. M. Srivastava

J. Math. Phys. 23, 357 (1982); http://dx.doi.org/10.1063/1.525375 (1 page)

Full Text: | Download PDF

Show Abstract
It 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.
Show PACS
02.30.-f Function theory, analysis
02.30.Jr Partial differential equations

The bi‐Hamiltonian structure of some nonlinear fifth‐ and seventh‐order differential equations and recursion formulas for their symmetries and conserved covariants

Benno Fuchssteiner and Walter Oevel

J. Math. Phys. 23, 358 (1982); http://dx.doi.org/10.1063/1.525376 (6 pages) | Cited 36 times

Full Text: | Download PDF

Show Abstract
Using 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.
Show PACS
02.30.Jr Partial differential equations

Singular symmetries of integrable curves and surfaces

B. A. Kupershmidt

J. Math. Phys. 23, 364 (1982); http://dx.doi.org/10.1063/1.525377 (3 pages) | Cited 1 time

Full Text: | Download PDF

Show Abstract
If w = w(x,t) is a solution of wt = 6wwx−wxxx+6ϵ2w2wx then ? = −w−ϵ−2 is also a solution. In general, integrable families and their members admit discrete symmetries whereas original systems may not.
Show PACS
02.30.Jr Partial differential equations
02.30.Uu Integral transforms
02.30.Vv Operational calculus

Recurrence relations for the coefficients of perturbation expansions

H. J. W. Müller‐Kirsten and L. K. Sharma

J. Math. Phys. 23, 367 (1982); http://dx.doi.org/10.1063/1.525378 (6 pages) | Cited 4 times

Full Text: | Download PDF

Show Abstract
Exact 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.
Show PACS
02.30.Lt Sequences, series, and summability
02.30.Mv Approximations and expansions

An identity in Riemann–Cartan geometry

H. T. Nieh and M. L. Yan

J. Math. Phys. 23, 373 (1982); http://dx.doi.org/10.1063/1.525379 (2 pages) | Cited 53 times

Full Text: | Download PDF

Show Abstract
We derive a new Gauss–Bonnet type identity in Riemann‐Cartan geometry: (−g)1/2ϵμνλρ (Rμνλρ + (1/2) CαμνCαλν) = ∂μ (−(−g)1/2ϵμνλρCμνλρ), where Cαμν is the torsion tensor.
Show PACS
02.40.Ky Riemannian geometries

Theta functions, Gaussian series, and spatially periodic solutions of the Korteweg–de Vries equation

John P. Boyd

J. Math. Phys. 23, 375 (1982); http://dx.doi.org/10.1063/1.525380 (13 pages) | Cited 18 times

Full Text: | Download PDF

Show Abstract
It has been shown by Novikov [Funct. Anal. Appl. 8, 236 (1974)], Dubrovin et al. [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.
Show PACS
02.60.-x Numerical approximation and analysis

String mechanics based on 2‐forms

P. Mitra

J. Math. Phys. 23, 388 (1982); http://dx.doi.org/10.1063/1.525381 (4 pages) | Cited 1 time

Full Text: | Download PDF

Show Abstract
Nambu invented a mechanics for strings by replacing the fundamental 1‐form pidqi−H dt 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.
Show PACS
46.05.+b General theory of continuum mechanics of solids

Sine‐Gordon and modified Korteweg–de Vries charges

K. M. Case and A. M. Roos

J. Math. Phys. 23, 392 (1982); http://dx.doi.org/10.1063/1.525382 (4 pages) | Cited 8 times

Full Text: | Download PDF

Show Abstract
The 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.
Show PACS
46.05.+b General theory of continuum mechanics of solids
03.65.Fd Algebraic methods

Inverse problems for nonabsorbing media with discontinuous material properties

Robert J. Krueger

J. Math. Phys. 23, 396 (1982); http://dx.doi.org/10.1063/1.525358 (9 pages) | Cited 21 times

Full Text: | Download PDF

Show Abstract
One‐dimensional electromagnetic and elastic inverse 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.
Show PACS
46.25.Cc Theoretical studies
03.50.De Classical electromagnetism, Maxwell equations

Feynman path integral and Poisson processes with piecewise classical paths

Ph. Combe, R. Hoegh‐Krohn, R. Rodriguez, M. Sirugue, and M. Sirugue‐Collin

J. Math. Phys. 23, 405 (1982); http://dx.doi.org/10.1063/1.525359 (7 pages) | Cited 6 times

Full Text: | Download PDF

Show Abstract
We prove the existence of a Feynman integral formula for gentle perturbations of the harmonic oscillator. This result is extended to Bose relativistic theory.
Show PACS
03.65.Db Functional analytical methods
03.65.Fd Algebraic methods

Theoretical basis for Coulomb matrix elements in the oscillator representation

Augustine C. Chen

J. Math. Phys. 23, 412 (1982); http://dx.doi.org/10.1063/1.525360 (5 pages) | Cited 17 times

Full Text: | Download PDF

Show Abstract
Hydrogenic 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.
Show PACS
03.65.Ge Solutions of wave equations: bound states

Simple multiple explode–decay mode solutions of a two‐dimensional nonlinear Schrödinger equation

Akira Nakamura

J. Math. Phys. 23, 417 (1982); http://dx.doi.org/10.1063/1.525361 (4 pages) | Cited 2 times

Full Text: | Download PDF

Show Abstract
Multiple 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.’’
Show PACS
03.65.Ge Solutions of wave equations: bound states

A combinatoric result related to the N‐body problem

Bengt R. Karlsson

J. Math. Phys. 23, 421 (1982); http://dx.doi.org/10.1063/1.525362 (2 pages) | Cited 2 times

Full Text: | Download PDF

Show Abstract
In 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 μN is shown to equal the Euler number ‖EN−1‖ if N is odd, and 2N(2N−1)‖BN‖/N, where BN 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, μN is the basic number of coupled equations when all particles are identical.
Show PACS
03.65.Nk Scattering theory
02.10.-v Logic, set theory, and algebra

A generalization of the Dirac equation to accelerating reference frames

John R. Urani and Marilyn H. Kemp

J. Math. Phys. 23, 423 (1982); http://dx.doi.org/10.1063/1.525363 (2 pages) | Cited 2 times

Full Text: | Download PDF

Show Abstract
Using a recently developed global isometry method for treating accelerating observers, the induced tangent space transformation on flat Lorentzian R4 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 C2 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.
Show PACS
04.20.-q Classical general relativity

The stationary coordinate systems in flat spacetime

John R. Letaw and Jonathan D. Pfautsch

J. Math. Phys. 23, 425 (1982); http://dx.doi.org/10.1063/1.525364 (7 pages) | Cited 22 times

Full Text: | Download PDF

Show Abstract
The 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.
Show PACS
04.20.Cv Fundamental problems and general formalism

Singular boundaries of space–times

Robert Geroch, Liang Can‐bin, and Robert M. Wald

J. Math. Phys. 23, 432 (1982); http://dx.doi.org/10.1063/1.525365 (4 pages) | Cited 15 times

Full Text: | Download PDF

Show Abstract
We 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 T1‐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.
Show PACS
04.20.Cv Fundamental problems and general formalism

The Riemann tensor, the metric tensor, and curvature collineations in general relativity

C. B. G. McIntosh and W. D. Halford

J. Math. Phys. 23, 436 (1982); http://dx.doi.org/10.1063/1.525366 (6 pages) | Cited 16 times

Full Text: | Download PDF

Show Abstract
The equation xμνRμ λαβ+xμλRμ ναβ = 0, where xμν and Rμ ναβ are the components of an arbitrary symmetric tensor and of the Riemann tensor formed from the metric tensor gμν, is trivially satisfied by xμν = ϕgμν. 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 gμν, from a given set of Rμ ναβ. We have found all Rμ ναβ for which there exist nontrivial solutions of the above equation, and we have given the form of the xμν in each case. Various examples of space–times for explicit nontrivial solutions are discussed.
Show PACS
04.20.Jb Exact solutions
04.20.-q Classical general relativity
02.40.Ky Riemannian geometries

Quantization of spinor fields. II. Meaning of ’’bosonization’’ in 1+1 and 1+3 dimensions

Piotr Garbaczewski

J. Math. Phys. 23, 442 (1982); http://dx.doi.org/10.1063/1.525367 (9 pages) | Cited 8 times

Full Text: | Download PDF

Show Abstract
We demonstrate that the correspondence principle allowing us to relate the classical (c number) and quantum levels of spinor fields in 1+1 and 1+3 dimensions, involves free Bose systems with unbounded from below Hamiltonians. The necessary condition for the quantum spinor fields to be ’’bosonized’’ on the ’’physical’’ space is that for the related free Bose systems, only the non‐negative part of the spectrum persists, due to constraints. Compared with the bosonization formulas, the number of independent Bose degrees of freedom necessary for a consistent formulation of the correspondence principle is doubled.
Show PACS
11.10.Ef Lagrangian and Hamiltonian approach

Reduction of the super phase space for a massless Dirac particle

Bengt E. W. Nilsson

J. Math. Phys. 23, 451 (1982); http://dx.doi.org/10.1063/1.525368 (5 pages)

Full Text: | Download PDF

Show Abstract
We show how to get the reduced super phase space for a classical spinning particle and how to quantize this theory. This technique introduces a Grassmannian ’’Hamiltonian’’ and a corresponding Grassmannian ’’time’’ parameter.
Show PACS
11.10.Ef Lagrangian and Hamiltonian approach
11.30.Pb Supersymmetry

Connection between the infinite sequence of Lie–Bäcklund symmetries of the Korteweg–de Vries and sine‐Gordon equations

P. Kaliappan and M. Lakshmanan

J. Math. Phys. 23, 456 (1982); http://dx.doi.org/10.1063/1.525369 (4 pages) | Cited 7 times

Full Text: | Download PDF

Show Abstract
From the observation that the infinite sequence of Lie–Bäcklund symmetries of the potential modified Kortweg–deVries (PMK–dV) and the sine–Gordon (s–G) equations are identical, it is shown that there exists a simple connection between the Lie–Bäcklund symmetries (written in the form of evolution equations) of the Korteweg–deVries (K–dV) and s–G equations. Further, this connection is similar to the one obtained by Chodos for the conserved quantities of K–dV and s–G equations. We also point out that the result of Chodos can be realized from the equality of conserved densities of PMK–dV and s–G systems.
Show PACS
11.30.-j Symmetry and conservation laws
02.30.Jr Partial differential equations
03.50.-z Classical field theories

Complementary energy principles in dissipative fluids

E. W. Laedke and K. H. Spatschek

J. Math. Phys. 23, 460 (1982); http://dx.doi.org/10.1063/1.525370 (4 pages) | Cited 5 times

Full Text: | Download PDF

Show Abstract
Necessary and sufficient conditions for exponential stability of equilibria in dissipative fluids are discussed on the basis of energy principles. First, a known maximum principle is reformulated in a manner which is more appropriate for evaluation of actual growth rates. Secondly, complementary variational principles are presented. The latter are quite useful for qualitative estimates and numerical computations since they lead to upper and lower bounds for the exact growth rates. The results are applicable to ideal magnetohydrodynamic (MHD) cases as well as resistive plasmas and also solitary waves.
Show PACS
47.20.-k Flow instabilities
47.10.-g General theory in fluid dynamics
52.30.-q Plasma dynamics and flow
52.35.Py Macroinstabilities (hydromagnetic, e.g., kink, fire-hose, mirror, ballooning, tearing, trapped-particle, flute, Rayleigh-Taylor, etc.)

Long‐range order in the spin van der Waals model

M. Howard Lee

J. Math. Phys. 23, 464 (1982); http://dx.doi.org/10.1063/1.525371 (8 pages) | Cited 18 times

Full Text: | Download PDF

Show Abstract
Long‐range order in the spin van der Waals model is considered when the number of spins is finite and also when infinite. We show explicitly that a finite system cannot support long‐range order. An infinite system at high temperatures is found to be dominated by the entropy of degenerate states of the system and, as a result, the system behaves essentially like an ideal system. In an infinite system at low temperatures, long‐range order exists, fully reflecting the spin symmetry of the Hamiltonian. For the XY‐like regime (J≳Jz ), mx is finite but mz vanishes, where mx and mz are reduced order parameters for the transverse and longitudinal directions, respectively. For the Ising‐like regime (J<Jz), mx vanishes but mz is finite. The isotropic interaction (Jz = J) behaves as a singularity and it must be considered separately. A physical interpretation of the behavior of long‐range order is offered using the geometry of spin space.
Show PACS
75.10.-b General theory and models of magnetic ordering
75.10.Dg Crystal-field theory and spin Hamiltonians
05.70.Jk Critical point phenomena
Close

Google Calendar

ADVERTISEMENT

Go to AIP Home   © AIP Publishing LLC
close