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Dec 1981

Volume 22, Issue 12, pp. 2727-3010

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Numerical representation and identification of graphs

H. H. Chen and Felix Lee

J. Math. Phys. 22, 2727 (1981); http://dx.doi.org/10.1063/1.525176 (5 pages) | Cited 2 times

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A method to represent each linear graph by a single number, the determinant of its modified incidence matrix, is introduced. The isomorphism of graphs can be determined by comparing the determinants of their incidence matrices. Although it is not proved that different graphs can always be distinguished by the determinants of their modified incidence matrices, the proposed method provides a good practical algorithm for the identification of graphs. Applications of the single‐number representation of graphs are discussed.
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02.10.De Algebraic structures and number theory
02.70.-c Computational techniques; simulations

Sp(6) states in an SU(3)×U(1) basis

R. Gaskell, G. Rosensteel, and R. T. Sharp

J. Math. Phys. 22, 2732 (1981); http://dx.doi.org/10.1063/1.525177 (4 pages) | Cited 8 times

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We find all missing label operators and also a complete set of analytic nonorthonormal basis states for the group–subgroup Sp(6)⊇SU(3)×U(1), both for the compact version of Sp(6) and for the noncompact Sp(6,R) relevant to the symplectic nuclear collective model.
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02.20.-a Group theory
21.60.Ev Collective models
21.60.Fw Models based on group theory

Generating functions for G2 characters and subgroup branching rules

R. Gaskell and R. T. Sharp

J. Math. Phys. 22, 2736 (1981); http://dx.doi.org/10.1063/1.525178 (4 pages) | Cited 15 times

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The G2 character generator is given; with its help generating functions are derived for branching rules for G2 irreducible representations reduced according to its maximal semisimple subgroups.
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02.20.Hj Classical groups

The global symmetries of spin systems defined on abelian groups. I

M. Marcu, A. Regev, and V. Rittenberg

J. Math. Phys. 22, 2740 (1981); http://dx.doi.org/10.1063/1.525179 (13 pages) | Cited 16 times

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We consider the classification problem of the global symmetry groups of spin systems defined on abelian groups. Its implications on the generating functional, the transfer matrix, the Hamiltonian formalism, and factorization properties of spin systems are discussed. The duality properties of spin systems defined on semidirect products of abelian groups are revisited. In the first of this series of three papers we list the groups for systems defined on Zp (p prime), Z2⊗Z2, and Z2⊗Z2⊗Z2 manifolds. They are direct or wreath products of M‐metacyclic groups and symmetric groups.
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02.20.Bb General structures of groups

The global symmetries of spin systems defined on abelian groups. II

M. Marcu and V. Rittenberg

J. Math. Phys. 22, 2753 (1981); http://dx.doi.org/10.1063/1.525180 (6 pages) | Cited 5 times

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We present the classification of the global symmetry groups of spin systems defined on Zp⊗Zq,Zp2, and Zp⊗Zp abelian groups ( p and q are prime numbers).
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02.20.Bb General structures of groups

On generalized torsion tensor fields and the reduced fiber bundle

Roberto Cianci

J. Math. Phys. 22, 2759 (1981); http://dx.doi.org/10.1063/1.525181 (3 pages) | Cited 4 times

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The reduction of a connection form ρ from a fiber bundle P to a subbundle Q is examined in detail; defining generalized torsion forms, we show how the usual Maurer–Cartan structural equations have to be modified. Examples and applications to classical general relativity and gauge theories are outlined.
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02.20.Qs General properties, structure, and representation of Lie groups

Representations of the groups GL(n,R) and SU(n) in an SO(n) basis

Bruno Gruber and Anatoli U. Klimyk

J. Math. Phys. 22, 2762 (1981); http://dx.doi.org/10.1063/1.525182 (8 pages) | Cited 6 times

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The explicit form for the infinitesimal operators and the (finite) matrix elements with respect to an SO(n) basis is obtained for the representations of the most degenerate series of the group SL(n,R), and for the irreducible unitary representations of the group SU(n) with highest weight (M,0,...,0).
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02.20.Qs General properties, structure, and representation of Lie groups

Second and fourth indices of plethysms

J. McKay, J. Patera, and R. T. Sharp

J. Math. Phys. 22, 2770 (1981); http://dx.doi.org/10.1063/1.525183 (5 pages) | Cited 10 times

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The direct product of several copies of a representation decomposes into a direct sum of components each with a definite permutation symmetry. The decomposition of any of the components into a direct sum of irreducible representations is the computation of a plethysm. The decomposition is often simply effected when the dimension and its analogs, the second and fourth indices of the plethysm, are known. The paper contains formulas for second and fourth indices of many specific plethysms as well as a prescription for the general plethysm. The same formula is valid for the plethysm based on any finite representation of any semisimple Lie algebra. Applications are illustrated by decomposition of all plethysms of degree 3 based on the E8 representation of dimension 3875; all fourth‐degree E8‐scalars are enumerated.
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02.20.Qs General properties, structure, and representation of Lie groups

Irreducible representations of the central extension of Sl(2) ΛT2

J. W. B. Hughes

J. Math. Phys. 22, 2775 (1981); http://dx.doi.org/10.1063/1.525184 (5 pages) | Cited 3 times

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Using shift operator techniques a classification is given of the irreducible star representations of the central extension algebra C(Sl(2)ΛT2). It is found to possess two generic series of such representations, together with an isolated representation which is just the metaplectic representation of Sl(2). This is the only representation it possesses in common with the superalgebra Osp(2, 1).
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02.20.Rt Discrete subgroups of Lie groups
02.20.Sv Lie algebras of Lie groups
02.10.Ud Linear algebra
02.10.Xm Multilinear algebra

Non‐self‐adjoint Zakharov–Shabat operator with a potential of the finite asymptotic values. I. Direct spectral and scattering problems

Naruyoshi Asano and Yusuke Kato

J. Math. Phys. 22, 2780 (1981); http://dx.doi.org/10.1063/1.525185 (14 pages) | Cited 14 times

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The Zakharov and Shabat equation for the scattering problem is studied: The estimates, analytical properties, and asymptotic expansions of the Jost solution are presented for a general class of the potentials Q(x) not vanishing at infinity. The existence of the similarity transformation is also shown. For Q(x) vanishing at infinity, the continuous part of the spectrum doubly degenerates. However, nonvanishing (finite) asymptotic values of Q(x) dissolve the degeneracy completely. The expansion theorem is given in C02(R) and for a class of Q(x) we prove that the Zakharov and Shabat equation yields a non‐self‐adjoint spectral operator in the Hilbert space in the sense of Dunford and Schwartz.
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02.30.Hq Ordinary differential equations
03.65.Nk Scattering theory
03.65.Db Functional analytical methods

Sensitivity theory for nonlinear systems. I. Nonlinear functional analysis approach

Dan G. Cacuci

J. Math. Phys. 22, 2794 (1981); http://dx.doi.org/10.1063/1.525186 (9 pages) | Cited 73 times

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Concepts of nonlinear functional analysis are employed to investigate the mathematical foundations underlying sensitivity theory. This makes it possible not only to ascertain the limitations inherent in existing analytical approaches to sensitivity analysis, but also to rigorously formulate a considerably more general sensitivity theory for physical problems characterized by systems of nonlinear equations and by nonlinear functionals as responses. Two alternative formalisms, labeled the ’’forward sensitivity formalism’’ and the ’’adjoint sensitivity formalism,’’ are developed in order to evaluate the sensitivity of the response to variations in the system parameters. The forward sensitivity formalism is formulated in normed linear spaces, and the existence of the Gâteaux differentials of the operators appearing in the problem is shown to be both necessary and sufficient for its validity. This formalism is conceptually straightforward and can be advantageously used to assess the effects of relatively few parameter alterations on many responses. On the other hand, for problems involving many parameter alterations or a large data base and comparatively few functional‐type responses, the alternative adjoint sensitivity formalism is computationally more economical. However, it is shown that this formalism can be developed only under conditions that are more restrictive than those underlying the validity of the forward sensitivity formalism. In particular, the requirement that operators acting on the state vector and on the system parameters must admit densely defined Gâteaux derivatives is shown to be of fundamental importance for the validity of this formalism. The present analysis significantly extends the scope of sensitivity theory and provides a basis for still further generalizations.
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02.30.Sa Functional analysis

Sensitivity theory for nonlinear systems. II. Extensions to additional classes of responses

Dan G. Cacuci

J. Math. Phys. 22, 2803 (1981); http://dx.doi.org/10.1063/1.524870 (10 pages) | Cited 44 times

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This work extends a recent, functional‐analytic formulation of sensitivity theory to include treatment of additional types of responses. There are physical systems where a critical point of a function that depends on the system’s state vector and parameters defines the location in phase‐space where the response functional is evaluated. The Gâteaux differentials giving the sensitivities of both the functional and the critical point to changes in the system’s parameters are obtained by alternative formalisms. The foward sensitivity formalism is the simpler and more general, but may be prohibitively expensive for problems with large data bases. The adjoint sensitivity formalism, although less generally applicable and requiring several adjoint calculations, is likely to be the only practical approach. Sensitivity theory is also extended to include treatment of general operators, acting on the system’s state vector and parameters, as response. In this case, the forward sensitivity formalism is the same as for functional responses, but the adjoint sensitivity formalism is considerably different. The adjoint sensitivity formalism requires expanding the indirect effect term, an element of a Hilbert space, in terms of elements of an orthonormal basis. Since as many calculations of adjoint functions are required as there are nonzero terms in this expansion, careful consideration of truncating the expansion is needed to assess the advantages of the adjoint sensitivity formalism over the forward sensitivity formalism.
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02.30.Sa Functional analysis

Resonance poles and Gamow vectors in the rigged Hilbert space formulation of quantum mechanics

A. Böhm

J. Math. Phys. 22, 2813 (1981); http://dx.doi.org/10.1063/1.524871 (11 pages) | Cited 64 times

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After a summary of the Rigged Hilbert space formulation of quantum mechanics and a brief statement of its advantages over von Neumann’s formulation, a mathematically correct definition of Gamow’s exponentially decaying vectors as generalized energy eigenvectors is suggested. It is shown that exponentially decaying vectors are obtained from the S‐matrix poles in the lower half of the second sheet and exponentially growing vectors from the S‐matrix poles in the upper half of the second sheet. Decaying ’’state’’ vectors are defined as functionals over half of the space of physical states and growing ’’state’’ vectors are defined as functionals over the other half. On functionals over these subspaces, the dynamical group of time development splits into two semigroups, one for t ≳ 0 and the other for t < 0. The generalized basis system connected with the spectrum of the Hamiltonian is transformed into a new basis system in which the exponentially decaying component of the density matrix is separated.
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03.65.Ca Formalism
23.60.+e α decay
11.55.-m S-matrix theory; analytic structure of amplitudes

Perturbed Hamiltonian systems

K. M. Case and A. M. Roos

J. Math. Phys. 22, 2824 (1981); http://dx.doi.org/10.1063/1.524872 (7 pages) | Cited 2 times

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It is shown that when a completely integrable Hamiltonian system is perturbed about a particular solution the resulting equations to all orders are completely integrable Hamiltonian systems. Numerous examples are worked out and some new constants for the original system are obtained.
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45.05.+x General theory of classical mechanics of discrete systems
02.30.Hq Ordinary differential equations

On the covariant differential of spin direction in the Finslerian deformation theory of ferromagnetic substances

Satoshi Ikeda

J. Math. Phys. 22, 2831 (1981); http://dx.doi.org/10.1063/1.524873 (4 pages) | Cited 1 time

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In the Finslerian deformation theory of ferromagnetic substances, each point (x) is endowed with the unit vector ( y) called the spin and the line‐element (x,y) is taken as the independent variable. The length of y is normalized at each point, so that the direction of y alone is noticed. This is the so‐called spin direction. In the case of the magnetization state, each vector y rotates to become parallel, in a Euclidean sense (not a Finslerian sense), to the direction of an applied magnetic field and the magnetostriction occurs there. Within the framework of Finsler geometry, this Euclidean ’’parallelism’’ of y cannot be grasped by the ordinary covariant differential of y (i.e., Dy), so that a new one (i.e., δy) must be introduced, which is nothing but the covariant differential of spin direction. Up to now, however, the geometrical meaning of δy and the relation between δy and Dy have not yet been clarified, so that these problems will be considered in this paper.
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46.05.+b General theory of continuum mechanics of solids
03.50.Kk Other special classical field theories

Signals and discontinuities in general relativistic nonlinear electrodynamics

Sabás Alarcón Gutiérrez, Alan L. Dudley, and Jerzy F. Plebañski

J. Math. Phys. 22, 2835 (1981); http://dx.doi.org/10.1063/1.524874 (14 pages) | Cited 22 times

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A theory of nonlinear electrodynamics in an arbitrary curved space–time is developed from the fundamental action functional for a charged perfect fluid. The equations for small perturbations on a fixed nonlinear background are then the initial point for a comprehensive study of the characteristic surfaces. The essential distinctions between linear and nonlinear electrodynamic interactions under the influence of gravitation are exhibited. Discontinuities in the first derivatives of small perturbations are encountered (1) which may be of general algebraic types for both the electrodynamic and gravitational fields and (2) which may have spacelike propagation. A specific set of constraints which would permit the propagation of these extraordinary radiative fronts is presented. If the physical organization of a particular problem is presumed to be sufficiently sensitive to the nonlinear nature of the dynamical interactions, then the application of traditional causal concepts may be unreliable when intuition derived from Maxwellian electrodynamics with noninteracting photons is anticipated to provide event horizons.
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03.50.Kk Other special classical field theories
04.20.Fy Canonical formalism, Lagrangians, and variational principles

Partitioning lower bounds for Bubnov–Galerkin’s eigenvalues

M. A. Abdel‐Raouf

J. Math. Phys. 22, 2849 (1981); http://dx.doi.org/10.1063/1.524875 (5 pages) | Cited 3 times

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Löwdin’s partitioning technique is extended for calculating energy‐lower bounds in Bubnov–Galerkin’s eigenvalue problems.
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03.65.Ca Formalism

Construction of (JJ∗)−1 in the Chandler–Gibson reaction theory

Wayne Polyzou

J. Math. Phys. 22, 2854 (1981); http://dx.doi.org/10.1063/1.525166 (4 pages)

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We exhibit a technique to construct formally the operators (JJ∗)−1 and (JΠJ∗)−1 in the Chandler–Gibson theory. Our construction is based on integral equations whose kernels can be made contractive with an appropriate choice of parameters. We discuss uniqueness and give representations of the solutions as uniformly convergent series.
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03.65.Ge Solutions of wave equations: bound states

Faddeev’s equations in differential form: Completeness of physical and spurious solutions and spectral properties

J. W. Evans and D. K. Hoffman

J. Math. Phys. 22, 2858 (1981); http://dx.doi.org/10.1063/1.525167 (14 pages) | Cited 17 times

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Faddeev type equations are considered in differential form as eigenvalue equations for non‐self‐adjoint channel space (matrix) Hamiltonians HF. For these equations in both the spatially confined and infinite systems, the nature of the spurious (nonphysical) solutions is obvious. Typically, these together with the physical solutions (given extra technical assumptions) generate a regular biorthogonal system for the channel space. This property may be used to provide an explicit functional calculus for the then real eigenvalue scalar spectral HF, to show that ±iHF generate uniformly bounded C0 semigroups and to simply relate HF to self‐adjoint Hamiltonian‐like operators. These results extend to the four‐channel Faddeev type equations where the breakup channel is included explicitly.
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03.65.Nk Scattering theory

Quantum‐mechanical scattering by impenetrable periodic surfaces

A. W. Sáenz

J. Math. Phys. 22, 2872 (1981); http://dx.doi.org/10.1063/1.525168 (13 pages) | Cited 3 times

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In this paper, we investigate the existence and completeness of the wave operators W± = s‐limt→±∞  exp (itH) P exp (−itH0) corresponding to the quantum‐mechanical scattering of nonrelativistic particles by certain classes of impenetrable noncompact surfaces bounding domains Ω ⊆Rν (ν ⩾2) which contain a half‐space and are contained in another half‐space. Here, H0 is the usual negative (distributional) Laplacian−Δ in H0 = L2(Rν ), H is the negative Dirichlet Laplacian in H = L2(Ω), and P is an appropriate identification operator. Under these conditions, we prove by elementary methods that W± exist as partially isometric operators whose initial sets have a transparent physical meaning. Suppose now that the domain Ω ⊆Rν also has the periodicity property (x,xν)∊Ω→(x+l,xν)∊Ω when l ranges over a Bravais lattice in Rν−1 , where we write x∊Rν as (x,xν), with x∊Rν−1 and xν ∊R. Then (a) RanW± = Hscatt (H) and (b) W± are asymptotically complete, in the sense that H = Hscatt(H)⊕Hsurf(H). Here, Hscatt (H) and Hsurf (H) are suitably defined subspaces of scattering and surface states of H, respectively. Results (a) and (b) are proved by reducing the original scattering problem to a family of ’’scattering’’ problems in a periodicity cell of Ω, using direct‐integral methods, and by then using methods analogous to those of Lyford. The present work constitutes a rigorous foundation for the theory of scattering of low‐energy atomic beams by crystal surfaces, considered as impenetrable periodic barriers. Our methods should also be applicable to rigorous investigations of classical scattering by periodic surfaces with Dirichlet or Neumann boundary conditions.
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03.65.Nk Scattering theory

Radiative degrees of freedom of the gravitational field in exact general relativity

Abhay Ashtekar

J. Math. Phys. 22, 2885 (1981); http://dx.doi.org/10.1063/1.525169 (11 pages) | Cited 22 times

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The radiative degrees of freedom of the gravitational field are isolated by analyzing the structure available at null infinity, JIt is shown thay they are coded in certain equivalence classes {D} of connections; all information about gravitational radiation can be extracted from the curvature tensors of these connections directly on J without any reference to the interior of space–time. The space of classical vacua—i.e., of {D} with trivial curvature—is analyzed. It is shown that the quotient ST/T of the BMS supertranslation group by its translation subgroup acts simply and transitively on this space. The available structure is compared with that of gauge theories. Since the entire discussion can be carried out onJ without any reference to the interior, it suggests a new approach to quantum gravity. This approach will be presented in detail in a subsequent paper.
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04.20.Cv Fundamental problems and general formalism
04.60.-m Quantum gravity
04.30.-w Gravitational waves

On a completely symmetric choice of space–time coordinates

G. H. Derrick

J. Math. Phys. 22, 2896 (1981); http://dx.doi.org/10.1063/1.525170 (7 pages) | Cited 6 times

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It is shown that for a wide class of space–times the four coordinates may be chosen in a completely symmetric way. Such symmetric coordinates have the same causal nature in that they are all measured by the same type of prescription involving clocks and/or measuring rods. In these coordinate systems the metric tensor is invariant with respect to any permutation of the coordinate labels 0, 1, 2, 3.
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04.20.Cv Fundamental problems and general formalism

Memory function approach to nonlinear deterministic systems: An exact linear equation

Takeo Nishigori

J. Math. Phys. 22, 2903 (1981); http://dx.doi.org/10.1063/1.525171 (7 pages) | Cited 5 times

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A set of nonlinear evolution equations is cast into an exact linear non‐Markovian equation with the memory kernel reflecting the nonlinearity and coupling with irrelevant variables. The equation is deterministic in contrast to the generalized Langevin equation derived in a similar way. The solution to the nonlinear equations is expressed by a sum of exponential functions. A simple illustrative example is treated to show the effectiveness of the present approach.
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05.20.Dd Kinetic theory
02.90.+p Other topics in mathematical methods in physics (restricted to new topics in section 02)

On the gravitational phase transition in the Thomas–Fermi model

J. Messer

J. Math. Phys. 22, 2910 (1981); http://dx.doi.org/10.1063/1.525172 (8 pages) | Cited 12 times

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The first order phase transition in an infinite system of gravitating fermions is analyzed in the canonical ensemble. Except for the question of nonmonotonicity of the mass distribution as a function of the chemical potential, we give an analytical proof for the existence of the phase transition. A single phase region is shown to exist for temperatures high compared to the gravitational energy.
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05.30.Fk Fermion systems and electron gas
05.70.Fh Phase transitions: general studies
31.15.bt Statistical model calculations (including Thomas-Fermi and Thomas-Fermi-Dirac models)

The KMS condition and regularity for the ideal Bose gas

Ph. de Smedt and G. Stragier

J. Math. Phys. 22, 2918 (1981); http://dx.doi.org/10.1063/1.525173 (3 pages)

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We prove that the KMS condition for the infinite ideal Bose gas at any temperature 1/β and chemical potential μ⩽0 implies regularity under certain conditions on the test function space. Also, explicitly irregular solutions are constructed indicating the difference between the KMS condition and Gibbs states.
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05.30.Jp Boson systems
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