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Volume 17, Issue 6, June 1976

Irreducible tensor method for a chain SU_{2}⊃⋅⋅⋅⊃ G ^{″}⊃G′⊃G and molecular physics
View Description Hide DescriptionThree things are discussed in this note around two results in the theory of angular momentum. First, certain Clebsch–Gordan coefficients of a compact topological group, which has a representation generalizing the antisymmetric representation [1^{ n }] of S_{ n }, are shown to possess an interesting symmetry property. Second, for physical purpose this property is applied to the octahedral group O≈S_{4} and its double (i.e., covering) group O* (i.e., O*/S_{2}≈O) considered as subgroups of SO_{3} and SU_{2}, respectively. We take this opportunity for briefly connecting some papers on the representation theory of SU_{2} recently published in this journal with some previous works by the author on the Wigner–Racah algebra of a noncanonical chain SU_{2}⊃G. Third, the material is used for rationalizing the well‐known (j=1) ‐ and less known (j=1/2) ‐isomorphisms of molecular physics in terms of isoscalar factors for the chains SO_{3}⊂O and SU_{2}⊂O*, respectively.

The Weyl transform of distributions
View Description Hide DescriptionThe Weyl transform is defined rigorously on the twisted product algebra of c ^{∞} functions slowly increasing at ∞. The image set under the Weyl rule of correspondence is shown to contain linear unbounded operators. In particular, the momentum and the coordinate operators P and Q and the polynomial functions of P and Q are included.

Almost singular potentials
View Description Hide DescriptionExplicit solutions are constructed for the lowest bound states of the Schrödinger equation with an attractive potential that behaves typically as r ^{−2+ε} at the origin. The energy levels and wavefunctions, which depend on the small parameter ε in a nonanalytic way, show some interesting properties; and some relations between this model and aspects of elementary particle physics are noted.

On the structure of simple pseudo Lie algebras and their invariant bilinear forms
View Description Hide DescriptionBy definition simple pseudo Lie algebras do not contain any nontrivial ideal. We show that ’’graded simplicity’’ implies ’’simplicity’’ and discuss the uniqueness of invariant bilinear forms on a simple pseudo Lie algebra. A lot of examples of simple pseudo Lie algebras is given together with their invariant bilinear forms. Under certain general assumptions we derive that the Lie algebrag contained in a simple pseudo Lie algebraa is reductive. Assuming that g is reductive, we prove that the ’’adjoint representation of g in the odd subspace of a’’ is completely reducible with at most two irreducible components. Finally we show that the pseudo Lie algebras with nondegenerate ’’generalized Killing form’’ are direct products of simple pseudo Lie algebras.

Phase integral approximations for calculating energy bands
View Description Hide DescriptionThe phase integral approximation is modified to incorporate an energy‐dependent effective potential that manifests the energy band character of lattice potentials. The resultant phase integral approximation because of its inherent renormalization demonstrates superior validity. Numerical examples which compare this approximation with other methods substantiate the superior convergence of this modified phase integral approximation.

Linear boson transformation coefficients
View Description Hide DescriptionA simple expression for the coefficients which connect a Fock state containing an arbitrary number of quasiparticles with its tranformed state under a boson Bogoliubov transformation is obtained.

The Riemann solution and the inverse quantum mechanical problem
View Description Hide DescriptionSystematic use is made of the Riemann function to find the conditions for the existence of the kernel of the inverse problem at fixed value of the angular momentum. When the reference potential is the centrifugal one, only an l‐dependent condition on the potential must be required in the Marchenko case; in the Gel’fand–Levitan case, the condition is l‐independent. When the Coulomb potential is included in the reference potential, an exponential decrease of the potential is needed in both instances.

Partial differential matrix equations for the inverse problem of scattering theory
View Description Hide DescriptionSufficient conditions for the existence of a continuous translation operator are found in the case of a system of differential equations in which the matrix potential has the singularity of the centripetal term. The sufficient conditions are found in terms of moments of the nuclear potential. The method used employs the Riemann Green’s function. Threshold energies introduce a threshold energy dependence into the translation kernel and lead to a requirement of an exponential decrease for terms of the matrix potential.

Some new identities of the Clebsch–Gordan coefficients and representation functions of SO(2,1) and SO(4)
View Description Hide DescriptionA systematic derivation of various relations and identities among the Clebsch–Gordan coefficients and for the representation functions of SO(4) and SO(2,1), is given. These relations are essential in work involving the matrix elements of arbitrary group elements in higher noncompact groups such as O(4,2).

Massive and massless supersymmetry: Multiplet structure and unitary irreducible representations
View Description Hide DescriptionUIR’s of the supersymmetryalgebra for the massive and massless cases are analyzed covariantly (without the use of induced representations) in terms of their component spins. For the massive case normalized basis vectors ‖p ^{2}≳0, j _{0}; σ; p jλ〉 are constructed, where j _{0} is the ’’superspin’’ and σ is an additional quantum number serving to distinguish the different ‖p jλ〉, the constituent p ^{2}≳0, spin‐j UIR’s of the Poincaré group. For the massless case, normalized basis vectors ‖p ^{2}=0, λ_{0}; pλ〉 are similarly constructed, where λ_{0} is the ’’superhelicity.’’ Matrix elements of the supersymmetry generators, in these bases, are explicitly given. The ’’σ basis’’ is used to define weight diagrams for the massive UIR’s of supersymmetry, and their properties are briefly described. Eigenfunctions ω_{σ}(ϑ) are also defined, and their connection with the reduction of higher spin massive superfields Φ^{ J }(x,ϑ) is discussed. Finally, it is shown how gauge dependence necessarily arises with certain massless superfields. The massless scalar superfield, both gauge‐dependent and gauge‐independent, is discussed as an example.

Classic theory of direct intermembrane interaction
View Description Hide DescriptionThe theory of action‐at‐a‐distance interaction between objects of two dimensions is studied. The theory developed has some similarities to the corresponding theory of one‐dimensional objects (strings). The fundamental invariants of the theory are found. A particular interaction is studied, and it is found that the interaction between closed membranes takes place via a pseudoscalar field.

Excitation of horizontally polarized waves in critical‐coupling regions where the permittivity gradient approaches zero—Full wave solutions
View Description Hide DescriptionMaxwell’sequations for inhomogeneous isotropic media are transformed into coupled ordinary differential equations for the wave amplitudes. To facilitate the solutions of these equations for dielectric layers with critical coupling regions, a generalized WKB approach is used. To this end, sets of auxiliary functions that are solutions to the wave equation for homogeneous or linearly varying permittivity profiles are employed. By introducing an additional set of auxiliary, parabolic cylindrical functions, the generalized WKB approach is extended to obtain suitable solutions for critical coupling regions of the dielectric layer where the gradient of the permittivity profile also approaches zero. Expressions for the reflection and transmission coefficients and the characteristic surface impedance for an inhomogeneous dielectric layer are derived as functions of the transverse wavenumber. Realizability and reciprocity relationships are also derived.

Generalized molecular distribution functions for fluids and occupation probabilities for lattice gases
View Description Hide DescriptionThe generalized distribution functions σ_{ n }(ω) specify the probability of finding a subset of n molecules in a specified configuration and simultaneously a subvolume empty of all other molecules, whereas the conventional distribution functions ρ_{ n } specify only the configuration. The Mayer integral equationtheory for the ρ_{ n } is generalized assuming short‐range intermolecular forces to express both the ρ_{ n } and σ_{ n } as sums of integrals over the σ_{ n } and a kernel dependent on the forces. For the nearest‐neighbor lattice gas, certain of these relations are equivalent to those obtained by Widom and Van Leeuwen for the probabilities f _{ n } that an empty site is surrounded by n filled sites. The intimate relation of these relations, the Kirkwood–Salsburg integral equations, and the generalized distribution functions is thus displayed.

The Galilei group and its connected subgroups
View Description Hide DescriptionAll the connected subgroups of the Galilei group G—and of its central extension —are determined and classified up to a conjugation of G—respectively —and also up to an isomorphism. In order to construct these subgroups, some general properties on subalgebras of a given Lie algebra have been proved. It is interesting to note that the subgroups of are derived from the subgroups of G.

Wave operators for long‐range hard‐core potentials
View Description Hide DescriptionIn a previous paper, we have proposed a definition of wave operators for hard‐core potentials of long range from the standpoint of time‐dependent single‐channel quantum scattering theory. In the present note, we prove the existence of these operators for a large class of such potentials. The proof involves a combination of methods previously used to establich the existence of wave operators for short‐range hard‐core potentials and long‐range potentials without hard cores.

High frequency approximations for elastic surface waves propagating along cylinders of general cross section
View Description Hide DescriptionThe propagation of high frequency elasticsurface waves along the generators of a homogeneous isotropic cylinder of general cross section is considered. The boundary surface is stress‐free and the surface waves, or Rayleigh waves, are disturbances whose amplitudes decay rapidly with depth into the cylinder. An approximate equation, and a refined one, are derived which describe the high frequency behavior of the surface wave modes. These approximate equations lead to the asymptotic results derived earlier by Wilson and the author for the case of an open boundary curve for which the curvature attains its algebraic maximum at a single point, and in fact they permit a more complete analysis of the higher order modes. Moreover, the refined approximate equation describes the behavior of the surface wave modes in the transition region, at high frequencies, between the case of cross‐sectional boundary curves of nonconstant (and not ’’almost’’ constant) curvature, for which the modes are localized, and the case of constant curvature, for which they are not localized. Some particular examples are considered.

Derivation of an exact quantum‐mechanical formula for the second term in the activity expansion of the correlation function S (Q,ω)
View Description Hide DescriptionWe derive an exact expression for the second term in the activity expansion of S (Q,ω), the Fourier transform of Van Hove’s correlation function, for a quantum system of bosons or fermions (spin neglected). The results for both the self and distinct parts of this function are given in terms of standard two‐particle T matrices. The detailed‐balance condition and zeroth and first moment relations are confirmed. An expression for the second term in the activity expansion of S (Q), the structure factor, is obtained. The derivation follows directly from the definition of S (Q,ω) using formal operator techniques; in addition, we show that the c o n s e r v i n g T a p p r o x i m a t i o n for two‐particle Green’s functions yields the same result.

A class of conservative diffusion processes with delta function initial conditions
View Description Hide DescriptionA class of conservative one‐dimensional diffusion processes is discussed which satisfy a delta function initial condition. A similarity relationship exists between the space and time variables. This class contains as special cases three well‐known diffusion processes and another one that very recently became of interest in the theory of superradiant emission.

Subgroups of the Poincaré group and their invariants
View Description Hide DescriptionThe continuous subgroups of the Poincaré group, classified into conjugacy classes in a previous article, are here classified into isomorphism classes. For each isomorphism class of Lie subalgebras all invariants are found, with a distinction made between Casimir operators (polynomials in the generators), rational invariants (rational functions of the generators), and general invariants (irrational and transcendental functions of the generators). All results are summarized in tables. The meaning of nonpolynomial invariants is briefly discussed and illustrated in examples.

Invariants of real low dimension Lie algebras
View Description Hide DescriptionAll invariant functions of the group generators (generalized Casimir operators) are found for all real algebras of dimension up to five and for all real nilpotent algebras of dimension six.