Volume 26, Issue 10, October 1985
Index of content:

Commutation factors on generalized Lie algebras
View Description Hide DescriptionGeneralized Lie algebras or color algebras, as we shall call them, are described by an Abelian grading group Γ and a commutation factor ε defined on Γ. In this paper Γ is assumed to be finite. It is shown that color algebras with the pair 〈Γ,ε〉 can also be considered as color algebras with the different pair 〈Γ’,ε’〉 and that as a result a canonical pair 〈Γ_{ c },ε_{ c }〉 is possible. It is further shown that, in fact, a unique ‘‘minimal’’ 〈Γ_{ c },ε_{ c }〉 can be used for all algebras with the pair 〈Γ,ε〉.

On a labeling for point group harmonics. I. Cubic group
View Description Hide DescriptionExpressions for the Γ_{4}, Γ_{6}, and Γ_{8} representations are derived in terms of the Γ_{1} representations for neighboring angular momenta. Expressions for Γ_{5}, Γ_{7}, and again for Γ_{8} are derived in terms of the Γ_{2} representations. A third set of expressions for Γ_{8} is derived in terms of the Γ_{3} representations. With an arbitrary choice of orthonormal sets of the Γ_{1}, Γ_{2}, and Γ_{3} representations, orthonormal sets of all other kinds of representations are thus well defined and can be entirely labeled using the parent representations. All Clebsch–Gordan coefficients are expressed in terms of those between parent representations (and a few other ones). The Γ_{4} and Γ_{5} representations defined here are not those conventionally used, but they provide a simpler expression for the fictitious spin coefficients. Tables of the parent representations Γ_{1}, Γ_{2}, and Γ_{3} are given for quaternary and ternary axes of quantization. Using the usual Clebsch–Gordan coefficients of SU(2), these tables allow us to obtain any representation for integer or half‐integer angular momenta up to j=25.

On a labeling for point group harmonics. II. Icosahedral group
View Description Hide DescriptionThe expressions for the Γ_{2}, Γ_{5}, Γ_{6}, Γ_{8}, and Γ_{9} representations are given in terms of those of Γ_{1} representations for neighboring angular momenta. The coefficients of the Γ_{4} representations are expressed in terms of those of Γ_{7} representations. Therefore, with an arbitrary choice of orthonormal sets of Γ_{1}, Γ_{3}, and Γ_{7} representations, orthonormal sets of other kinds of representations are well defined and can be labeled with the labels of parent representations. All Clebsch–Gordan coefficients are expressed in terms of those between parent representations (and a few others). Tables of all nondegenerate Γ_{1}, Γ_{3}, and Γ_{7} representations are given, for the axes of quantization of order 5. With some degenerate Γ_{3} and Γ_{7} representations, which are also given, any representation of integer or half‐integer angular momentum up to 27 can be obtained using some usual Clebsch–Gordan coefficients of SU(2).

Algebra and physics of the unitary multiplicity‐free representations of ∼(SL)(4, R)
View Description Hide DescriptionThe systematics of the multiplicity‐free unitary irreducible representations of SL(4,R) are restudied, and an amended list is presented. An automorphism essential to the physical application for particles and fields in Minkowski space is described.

Character constraints on duality
View Description Hide DescriptionDuality identifies recoupling coefficients and isoscalar factors of the unitary groups with matrix elements of double coset representatives of the symmetric groups. When the double coset factorization is with respect to the same subgroup chain the matrix is that of the ordinary irreducible representation in a basis symmetry adapted to the subgroup chain. The invariant character of the symmetric group element then acts as a constraint on the magnitude, phase, and multiplicity resolution for diagonal elements. An algorithm for determining the content of this constraint is given and is shown to be consistent with a phase convention proposed previously.

Nonlinear evolution equations associated with a Riemann–Hilbert scattering problem
View Description Hide DescriptionIn an earlier paper nonlinear evolution equations associated with a Riemann–Hilbert scattering problem, which reduces, in an appropriate limit, to the Zakharov–Shabat–AKNS scattering problem, were considered. Here we discuss certain necessary constraints associated with the scattering problem and their impact upon the associated evolution equations. Moreover, the direct linearization of the nonlinear evolution equations and an algorithm to construct anN‐soliton solution are given.

Transforms associated to square integrable group representations. I. General results
View Description Hide DescriptionLet G be a locally compact group, which need not be unimodular. Let x→U(x) (x∈G) be an irreducible unitary representation of G in a Hilbert spaceH(U). Assume that U is square integrable, i.e., that there exists in H(U) at least one nonzero vector g such that ∫‖( U(x)g,g )‖^{2} d x<∞. We give here a reasonably self‐contained analysis of the correspondence associating to every vector f∈H(U) the function ( U(x)g,f ) on G, discussing its isometry, characterization of the range, inversion, and simplest interpolation properties. This correspondence underlies many properties of generalized coherent states.

On a computation formula for the representation matrices of U(n)
View Description Hide DescriptionA d‐matrix element of U(n), which is called the semi‐highest‐weight d‐matrix element and plays an essential role for the D‐matrix elements, is explicitly determined. By using the result, a formula of computing the D‐matrix elements of U(n) is given in terms of lowering operators corresponding to those of Nagel and Moshinsky.

Satake diagrams, Iwasawa decompositions, and representations of the exceptional Lie group F _{4}(−20)
View Description Hide DescriptionThe parabolic subgroups of the noncompact, exceptional Lie groupF _{4}(−20) are computed from a systematic analysis of Iwasawa and Langlands decompositions. Satake diagrams have been used to determine the involutive automorphisms of F _{4}(−20) which facilitate the Iwasawa decompositions. The polarizations associated with noncompact orbits with parabolic subgroups are computed. The representations so obtained for these polarizations using Kostant’s induction scheme yield the principal series representations of F _{4}(−20).

Thermodynamics of dimers on a rectangular L×M×N lattice
View Description Hide DescriptionThe exact closed‐form analytic solution of the problem of dimers on infinite two‐dimensional and three‐dimensional lattices is obtained. Entropy, isothermal compressibility, and constant pressure heat capacity of the system are given in terms of the normalized number density of dimers. The absolute activity of dimers is also given in terms of the normalized number density; it exhibits a behavior near close packing with a critical exponent exactly equal to 2, and with an amplitude 1/(4φ), where φ is the molecular freedom per dimer at close packing.

Solution of the eigenvalue problem of an integral equation with the help of its associated differential equation applied to the calculation of diffraction losses in confocal resonators
View Description Hide DescriptionA problem in optics is chosen in order to develop a general method for solving the eigenvalue problem of a homogeneous integral equation with the help of the eigenfunctions of the associated differential equation. The problem chosen is that of the modes of optical resonators with circular, confocal mirrors which are given by the solutions of a homogeneous Fredholm integral equation which can be derived from Kirchhoff ’s diffraction formula. This integral equation can be converted into a hyperspheroidal differential equation supplemented by appropriate boundary conditions. The solutions and eigenvalues of this differential equation are studied in detail for both small and large values of a parameter called the Fresnel number. These eigenfunctions are then used for the computation of the eigenvalues of the original integral equation which measure the diffraction loss in the resonator. Throughout the same general perturbation method is used, and our emphasis is on the solution of the general problem of solving the eigenvalue problem of the homogeneous integral equation together with that of its related differential equation.

First integrals for the modified Emden equation q̈+α(t) q̇+q ^{ n } =0
View Description Hide DescriptionIt is shown that the modified Emden equationq̈+α(t)q̇+q ^{ n }=0 possesses first integrals for functions α(t) other than k t ^{−} ^{1}. The function α(t) is obtained explicitly in the case n=3 and parametrically for other n(≠2). The case n=2 is seen to be particularly difficult to solve.

Compact quantum systems: Internal geometry of relativistic systems
View Description Hide DescriptionA generalization is presented of the kinematical algebra so(5), shown previously to be relevant for the description of the internal dynamics (Z i t t e r b e w e g u n g) of Dirac’s electron. The algebra so(n+2) is proposed for the case of a compact quantum system with n degrees of freedom. Associated wave equations follow from boosting these compact quantum systems. There exists a contraction to the kinematical algebra of a system with n degrees of freedom of the usual type, by which the commutation relations between n coordinate operators Q _{ i } and corresponding momentum operators P _{ i }, occurring within the so(n+2) algebra, go over into the usual canonical commutation relations. The so(n+2) algebra is contrasted with the sl(l,n) superalgebra introduced recently by Palev in a similar context: because so(n+2) has spinor representations, its use allows the possibility of interpreting the half‐integral spin in terms of the angular momentum of internal finite quantum systems. Connection is made with the ideas of Weyl on the possible use in quantum mechanics of ray representation of finite Abelian groups, and so also with other recent works on finite quantum systems. Possible directions of future research are indicated.

Deficiency indices and singular boundary conditions in quantum mechanics
View Description Hide DescriptionWe consider Schrödinger operators H in L ^{2}(R^{ n }), n ∈ N, with countably infinitely many local singularities of the potential which are separated from each other by a positive distance. It is proved that due to locality each singularity yields a separate contribution to the deficiency index of H. In the special case where the singularities are pointlike and the potential exhibits certain symmetries near these points we give an explicit construction of self‐adjoint boundary conditions.

On the semiclassical description of N‐level systems interacting with radiation fields
View Description Hide DescriptionIt is shown that the dynamics of an N‐level quantum system driven by a classical radiation field can be derived rigorously from an associated fully quantum‐mechanical model with the help of an asymptotic limit. By formally expanding the full quantum dynamics around the semiclassical limit, quantum corrections to the semiclassical description are systematically constructed. The observables describing these corrections are shown to exist in any order, and explicit expressions for their time dependence are given.

On the algebras of local observables in the generalized sense in quantum mechanics
View Description Hide DescriptionThe notion of observables localized with respect to a general spectral measure is introduced as a generalization of the localization with respect to the spectral measure of the position observable. The physical features of these observables are discussed and certain of their algebraic and lattice theoretic properties are presented.

On the connection between Schrödinger and Dirichlet forms
View Description Hide DescriptionRelations between Schrödinger forms associated with Schrödinger operators in L ^{2}(Ω;d ^{ n } x), Ω ⊂ R^{ n } open, n≥1 and the corresponding Dirichlet forms are investigated. Various concrete examples are presented.

Not necessary but sufficient condition for the positivity of generalized Wigner functions
View Description Hide DescriptionMany authors, both in quantum mechanics and signal theory, are concerned with a condition of positivity for the generalized Wigner functions. They give (and use) as necessary and sufficient the condition that the kernel be the characteristic function of another Wigner function. It is shown that the condition is not necessary, but only sufficient.

Spectral properties of the Dirac equation with anomalous magnetic moment
View Description Hide DescriptionThough the electron possesses an anomalous magnetic moment, this term has so far been treated by perturbation methods only. Here this rather singular term is treated nonperturbatively. For a general and large class of spherical potentials it is shown that the Hamiltonian is essentially self‐adjoint. If the potentials vanish at infinity the essential spectrum is R\(−m,m). These results hold for all, also strong, Coulomb‐type potentials. We further establish that the eigenvalues for such Hamiltonians are stable and can be computed by a low‐order perturbation theory, if l≥3. The above results can also be extended to multicenter potentials.

The relation between the (N )‐ and (N−1)‐electron atomic ground states
View Description Hide DescriptionThe relations between the ground states of an N‐ and (N−1)‐electron atomic system are studied. It is shown that in some directions of the configuration space, the ratio of the N‐electron atomic ground state to the one‐particle density is asymptotically equivalent to the (N−1)‐electron atomic ground state.