Index of content:
Volume 24, Issue 3, March 1983

A unified theory of the point groups. II. The general projective representations and their application to space groups
View Description Hide DescriptionThe general expressions of all inequivalent irreducible projective representations of crystallographic and noncrystallographic point groups (except for the icosahedral group) are given in terms of the vector representations of the proper double point groups. Its application to the representations of the space group of wave vector is discussed. Construction of the basis sets is discussed.

A unified theory of the point groups. III. Classification and basis functions of improper point groups
View Description Hide DescriptionThis paper introduces a new system of classification of improper point groups which is most effective for describing their general irreducible representations. The complete set of the general angular momentum eigenfunctions is classified according to the general irreducible representations of the seven sets of the improper point groups corresponding to C _{ n }(C _{∞}) and D _{ n }(D _{∞}) for an aribtrary n.

A unified theory of the point groups. IV. The general corepresentations of the crystallographic and noncrystallographic Shubnikov point groups
View Description Hide DescriptionA new system of classification for all the Shubnikov point groups G ^{ s } is presented, which is best suited for describing their isomorphisms as well as their construction. By the new classification and the modified form of the corepresentations, there are presented the simple general expressions of the irreducible corepresentations for the 10 minimum sets of proper G ^{ s }, from which all coirreps of the remaining G ^{ s } follow. (Icosahedral groups are excluded.)

Phase conventions consistent with duality between SUn and S _{ L }
View Description Hide DescriptionThe Schur–Weyl construction of irreducible tensors by symmetrized Lth rank exterior products of a defining n‐dimensional vector space establishes a duality between the coupling algebra of the unitary group Un and the symmetric group S _{ L } . The coupling coefficients shared by these two groups are real and possess symmetries due to complex conjugation in Un and association in S _{ L } . The unitary group is factored by its unitary unimodular subgroup SUn≂Un/U1 so it is usual to identify the coupling algebras of Un and SUn by choosing the trivial (phaseless) coupling for U1. This establishes a set of equivalence relations since all pseudoscalar irreducible representations (irreps) of Un subduce onto the scalar irrep of SUn. The question remains can this trivial identification with the coupling algebra of SUn be made consistent with the symmetries which follow from duality? Specifically the symmetries which follow from duality must be consistent with the equivalence relations. We argue such consistency may be obtained. The resulting symmetries place significant restrictions on the resolution of multiplicities and the choice of phase conventions for transformations which in the absence of duality are normally considered independently. We review the usual formulation of a coupling algebra and the identifications which follow from duality using a double coset decomposition of S _{ L } . Particular attention is given to the phase transformation introduced by a transposition in order of the two component irreps being coupled. Phase transformations needed for association and complex conjugation are examined to determine the symmetries they must exhibit to be consistent with duality. Two phase transformations needed for complex conjugation, the Derome–Sharp matrix and the 1j m factor, are shown to be related by association. A transposition phase convention and an association phase convention are proposed and shown to be consistent with all the symmetries required by duality.

The Racah algebra for groups with time reversal symmetry. III
View Description Hide DescriptionThe coupling of basis vectors and the recoupling of irreducible corepresentations is considered. The difference between the coupling coefficient and 3j mtensors leads to a variety of 6j and recoupling tensors.

On the symmetries of the 6j symbol
View Description Hide DescriptionThe 6jtensor for compact groups is shown to transform as a basis vector for the identity representation of the permutation group S _{4}. This allows character theory to be used to determine the minimum number of independent components and a projection operator to determine the relations between components—the symmetry properties.

Models of nonlinear representations and examples of linearization techniques
View Description Hide DescriptionExamples of classes of nonlinear representations of Lie groups are given. Nonlinear representations which are a perturbation of a unitary representation of the discrete series of SU(1,1) are then proved to be formally linearizable.

Euler angle parametrization of the complex rotation group and its subgroups. I
View Description Hide DescriptionA parametrization of the n‐dimensional complex rotation group O^{+}(n, C) by a set of (1/2)n(n−1) complex Euler angles is obtained in exactly the same way as that for the n‐dimensional real rotation group O^{+}(n, R). Certain subgroups L̂(n, r) of O^{+}(n, C) are then considered and found isomorphic with the ‘‘generalized Lorentz groups’’ L(n, r), of which the special case L(4, 1) is the usual Lorentz groupL of special relativity. Distinguishing features, in the form of reality nature, of Euler angles of L̂(n, r) are obtained for r=1, 2, and it is proved that these lead naturally to the definition of ‘‘real’’ Euler angles of the corresponding L(n, r), which, of course, include the Lorentz groupL.

Euler angle parametrization of the complex rotation group and its subgroups. II
View Description Hide DescriptionThe work of Paper I is extended here by obtaining the reality nature of Euler angles of the subgroups L̂(n,r) (defined there) of O^{+}(n,C), for arbitrary r, and then showing that these again enable one to define ‘‘real’’ Euler angles of the generalized Lorentz groupsL(n,r), r arbitrary.

The Clebsch–Gordan coefficients of the three‐dimensional Lorentz algebra in the parabolic basis
View Description Hide DescriptionStarting from the oscillator representation of the three‐dimensional Lorentzalgebra so(2,1), we build a Lie algebra of second‐order differential operators which realizes all series of self‐adjoint irreducible representations. The choice of the common self‐adjoint extention over a two‐chart function space determines whether they lead to single‐ or multivalued representations over the corresponding Lie group. The diagonal operator defining the basis is the parabolic subgroup generator. The direct product of two such algebras allows for the calculation of all Clebsch–Gordan coefficients explicitly, as solutions of Schrödinger equations for Pöschl–Teller potentials over one (D×D), two (D×C), or three (C×C) charts. All coefficients are given in terms of up to two _{2} F _{1} hypergeometric functions.

Upper and lower bounds for eigenvalues of nonlinear elliptic equations: I. The lowest eigenvalue
View Description Hide DescriptionWe give a method for finding bounds for the lowest eigenvalue of nonlinear elliptic equations with monotone, local, nonlinearities. This is an extension to nonlinear problems of Barta’s method for linear elliptic operators.

Evolution equations with high order conservation laws
View Description Hide DescriptionAn asymptotic algorithm presented in a previous paper is applied to investigate the possible structures of evolution equations u _{ t }=u _{ M }+K(u,...,u _{ M−1}), M=3,5, which could be compatible with the existence of a conserved density ρ_{0}(u), depending only on u, and with the existence as well of conserved densities with arbitrarily high‐order derivatives. For M=3 it is shown that the Calogero–Degasperis–Fokas equation is essentially the only nonpolynomial equation of that type. For the case u _{ t }=D[u _{4}+Q], with Q(u,...,u _{3}) a polynomial, we find a very narrow class of admissible structures for Q, typified by the few particular examples known up to date. Actually, there is in this case an essentially unique structure, modulo a Miura transformation.

Geometric construction and properties of some families of solutions of nonlinear partial differential equations. I
View Description Hide DescriptionThis series of papers deals with ‘‘The 19th Century Theory of Partial Differential Equations from an Advanced Standpoint.’’ In the treatises of Darboux, Goursat, and Forsythe one finds methods for classifying nonlinear differential equations according to the geometric properties of families of solutions. This work was used by Elie Cartan in his theory of exterior differential systems, and is involved in an indirect way in today’s work on ‘‘nonlinear physics.’’ I plan to present several major themes of the classical work (e.g., ‘‘general,’’ ‘‘singular,’’ and ‘‘complete’’ solutions, ‘‘intermediate integral’’) using geometric methods developed by Cartan, Vessiot, Ehresmann, and Spencer. My aim is to develop this material from a point of view that is both fundamental and directed toward its ultimate application. Another possible utilization is in the development of the symbolic computationcomputer systems such as MACSYMA and REDUCE. This first paper concentrates on a description of what is meant by a ‘‘general solution,’’ and presents the classical Lagrange–Charpit method for constructing ‘‘general’’ solutions by means of a ‘‘complete solution.’’ The Monge–Ampère equation is also treated in modernized form. A new geometric concept, that of a L a g r a n g e–V e s s i o t s u b m e r s i o n, is isolated from the classical literature.

The Painlevé property for partial differential equations
View Description Hide DescriptionIn this paper we define the Painlevé property for partial differential equations and show how it determines, in a remarkably simple manner, the integrability, the Bäcklund transforms, the linearizing transforms, and the Lax pairs of three well‐known partial differential equations (Burgers’ equation, KdV equation, and the modified KdV equation). This indicates that the Painlevé property may provide a unified description of integrable behavior in dynamical systems (ordinary and partial differential equations), while, at the same time, providing an efficient method for determining the integrability of particular systems.

Real axis integration of Sommerfeld integrals with applications to printed circuit antennas
View Description Hide DescriptionPrinted circuit antennas are becoming an integral part of imaging arrays in microwave, millimeter, and submillimeter wave frequencies. The electrical characteristics of such antennas can be analyzed by solving integral equations of the Fredholm first kind. The kernel involves Sommerfeld integrals which are particularly difficult to solve when source and field points lie on an electrical discontinuity, as it occurs in the determination of the characteristics of printed circuit antennas. An analytic‐numeric real axis integration technique has been developed for such integrals and it is combined with piece‐wise sinusoidal expansions to solve the Fredholm integral equation for the unknown current density.

A generalized Weyl correspondence. II. Some general results
View Description Hide DescriptionAfter defining a generalized Weyl correspondence, we give some general results. These are presented as comments on a theorem. They mainly refer to the finite‐dimensional, unbounded, and non‐self‐adjoint cases.

An upper bound on the allowed bands of the Bloch spectrum of one‐dimensional Schrödinger operators with periodic potentials
View Description Hide DescriptionUsing the principle of increasing flux and its complement, an upper bound associated with the allowed bands of the Bloch spectrum of one‐dimensional Schrödinger operators with periodic potentials is derived. Namely, let a be the period of the potential, E _{0} the lower bound of the Bloch spectrum and (E _{2( j−1)},E _{2j−1}) the allowed bands ( j=1,2,...). Then π>a[(E _{2j−1}−E _{0})^{1} ^{/} ^{2} −(E _{2(j−1)}−E _{0})^{1} ^{/} ^{2}].

Explicit Hilbert space representations of Schrödinger states: Definitions and properties of Stieltjes–Tchebycheff orbitals
View Description Hide DescriptionStieljes–Tchebycheff orbital approximations are described for the discrete and continuun Schrödinger states of self‐adjoint (Hamiltonian) operators, and certain of their properties are established and clarified. The nth order orbitals are defined in accordance with corresponding moment‐theory approximations to spectral densities as eigenfunctions of the appropriate operator in an n‐term Cauchy–Hilbert space. Their eigenvalues are consequently the generalized Gaussian or Radau quadrature points of the associated spectral density function formed by projection with an appropriate test function on the Schrödinger states, and their norms provide the corresponding (reciprocal) quadrature weights. Stable algorithms are described for their construction employing recursive Lanczos and orthogonal‐polynomial methods. In finite orders the spatial characteristics of the orbitals correspond to spectral averages in the neighborhoods of the quadrature points over the correct Schrödinger states. The spectral content of an individual orbital is obtained in closed form without reference to the correct underlying Schrödinger states. Convergence (n→∞) is obtained in the discrete spectral region to Schrödinger eigenstates of finite norm, whereas in the essential portion of the spectrum the orbitals converge to scattering states of improper (infinite) norm. Connections with matrix partitioning and optical potentialtheory are made, indicating that first‐ order orbitals can provide exact Schrödinger states over local predetermined portions of configuration space. Illustrative computational studies of the regular l‐wave spectra of the Coulomb Hamiltonian are provided.

On the correspondence between classical and quantum mechanics. I
View Description Hide DescriptionWe consider here a general procedure for implementing the correspondence between classical and quantum mechanical systems of finitely many degrees of freedom. We show that corresponding systems may be formulated within a common framework in such a way that the kinematic, statistical, dynamical, and covariance features may be easily compared.

A class of solvable Pauli–Schrödinger Hamiltonians
View Description Hide DescriptionThe Pauli–Schrödinger equation for a spin 1/2, neutral particle with a nonvanishing magnetic moment μ_{0}, interacting with an external scalar potential V and a static magnetic fieldB, both functions of only one of the coordinates, is solved exactly for four different choices of the potential and the field. By choosing in the examples the coordinate y, we present these solutions in the following cases: (i) V( y)=0, B( y)=(B _{0}sin κy, 0, B _{0}cos κy) where B _{0} and κ are two arbitrary constants. (ii) V( y)=λ′y ^{2}, B( y)=‖αy‖(cos 2κy, 0, sin 2κy) where 6* λ=ℏ^{2}/2m and α,κ arbitrary constants and λ′=λ(μ_{0}/2κλ)^{2}. (iii) V( y)=λ′(c̄_{1}tan αy+c̄_{2}cot αy)^{2}, B( y)=B( y) (sin (2κy+2δ(y)), 0, cos (2κy+2δ( y)) where B ^{2}( y)=(c̄_{1}tan αy+c̄_{2}cot αy)^{2} (+α^{2}/4κ^{2})(c̄_{1}sec^{2}αy−c̄_{2}csc^{2}αy)^{2} and tan 2δ( y)=(2κ/λ)((c̄_{1}tan αy+c̄_{2}cot αy) /(c̄_{1}sec^{2}αy−c̄_{2}csc^{2}αy)); c̄_{1}, c̄_{2}, α, and κ are arbitrary constants. (iv) V( y)=λ′(ā tanh z+b̄)^{2} where z =[(y−c d)/d], B ^{2}( y)=(ā tanh z+b̄)^{2}+(ā^{2}/4κ^{2} d ^{2})sech^{2} z, tan 2δ( y)=(2κd/ā)((ā tanh z+b̄)/sech^{2} z); ā, b̄, c, d, and κ are arbitrary constants restricted only by (μ_{0}/2κλ)ā^{2}+ā/d>0. The functions B( y) and δ( y) define the vector B( y) as in (iii). Our method of solution is based on the familiar factorization technique for solving the Schrödinger eigenvalue equation. Several interesting physical results of these solutions are discussed.