Volume 22, Issue 12, December 1981
Index of content:

Numerical representation and identification of graphs
View Description Hide DescriptionA 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.

Sp(6) states in an SU(3)×U(1) basis
View Description Hide DescriptionWe 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.

Generating functions for G _{2} characters and subgroup branching rules
View Description Hide DescriptionThe G _{2} character generator is given; with its help generating functions are derived for branching rules for G _{2} irreducible representations reduced according to its maximal semisimple subgroups.

The global symmetries of spin systems defined on abelian groups. I
View Description Hide DescriptionWe 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 Z _{ p } (p prime), Z _{2}⊗Z _{2}, and Z _{2}⊗Z _{2}⊗Z _{2}manifolds. They are direct or wreath products of M‐metacyclic groups and symmetric groups.

The global symmetries of spin systems defined on abelian groups. II
View Description Hide DescriptionWe present the classification of the global symmetry groups of spin systems defined on Z _{ p }⊗Z _{ q },Z _{ p 2 }, and Z _{ p }⊗Z _{ p } abelian groups ( p and q are prime numbers).

On generalized torsion tensor fields and the reduced fiber bundle
View Description Hide DescriptionThe 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.

Representations of the groups GL(n,R) and SU(n) in an SO(n) basis
View Description Hide DescriptionThe 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).

Second and fourth indices of plethysms
View Description Hide DescriptionThe 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 E _{8} representation of dimension 3875; all fourth‐degree E _{8}‐scalars are enumerated.

Irreducible representations of the central extension of Sl(2) ΛT_{2}
View Description Hide DescriptionUsing shift operator techniques a classification is given of the irreducible star representations of the central extension algebraC(Sl(2)ΛT_{2}). 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).

Non‐self‐adjoint Zakharov–Shabat operator with a potential of the finite asymptotic values. I. Direct spectral and scattering problems
View Description Hide DescriptionThe 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 C _{0} ^{2}(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.

Sensitivity theory for nonlinear systems. I. Nonlinear functional analysis approach
View Description Hide DescriptionConcepts 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.

Sensitivity theory for nonlinear systems. II. Extensions to additional classes of responses
View Description Hide DescriptionThis 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.

Resonance poles and Gamow vectors in the rigged Hilbert space formulation of quantum mechanics
View Description Hide DescriptionAfter 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.

Perturbed Hamiltonian systems
View Description Hide DescriptionIt 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.

On the covariant differential of spin direction in the Finslerian deformation theory of ferromagnetic substances
View Description Hide DescriptionIn 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., D y), 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 D y have not yet been clarified, so that these problems will be considered in this paper.

Signals and discontinuities in general relativistic nonlinear electrodynamics
View Description Hide DescriptionA 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.

Partitioning lower bounds for Bubnov–Galerkin’s eigenvalues
View Description Hide DescriptionLöwdin’s partitioning technique is extended for calculating energy‐lower bounds in Bubnov–Galerkin’s eigenvalue problems.

Construction of (J J*)^{−1} in the Chandler–Gibson reaction theory
View Description Hide DescriptionWe exhibit a technique to construct formally the operators (J J*)^{−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.

Faddeev’s equations in differential form: Completeness of physical and spurious solutions and spectral properties
View Description Hide DescriptionFaddeev type equations are considered in differential form as eigenvalueequations for non‐self‐adjoint channel space (matrix) Hamiltonians H_{ F }. 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 functionalcalculus for the then real eigenvalue scalar spectral H_{ F }, to show that ±iH_{ F }generate uniformly bounded C _{0} semigroups and to simply relate H_{ F } to self‐adjoint Hamiltonian‐like operators. These results extend to the four‐channel Faddeev type equations where the breakup channel is included explicitly.

Quantum‐mechanical scattering by impenetrable periodic surfaces
View Description Hide DescriptionIn this paper, we investigate the existence and completeness of the wave operators W _{±} = s‐lim_{ t→±∞} exp (i t H) P exp (−i t H _{0}) corresponding to the quantum‐mechanical scattering of nonrelativistic particles by certain classes of impenetrable noncompact surfaces bounding domains Ω ⊆R^{ν} (ν ⩾2) which contain a half‐space and are contained in another half‐space. Here, H _{0} is the usual negative (distributional) Laplacian−Δ in H_{0} = L ^{2}(R^{ν} ), H is the negative Dirichlet Laplacian in H = L ^{2}(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 ⊆R^{ν} also has the periodicity property (x̃,x _{ν})∈Ω→(x̃+l,x _{ν})∈Ω when l ranges over a Bravais lattice in R^{ν−1} , where we write x∈R^{ν} as (x̃,x _{ν}), with x̃∈R^{ν−1} and x _{ν} ∈R. Then (a) RanW _{±} = H_{scatt} (H) and (b) W _{±} are asymptotically complete, in the sense that H = H_{scatt}(H)⊕H_{surf}(H). Here, H_{scatt} (H) and H_{surf} (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.