Clebsch‐Gordan Coefficients and Special Function Identities. I. The Harmonic Oscillator Group
It is shown that by constructing explicit realizations of the Clebsch‐Gordan decomposition for tensor products of irreducible representations of a group G, one can derive a wide variety of special function identities with physical interest. In this paper, the representation theory of the harmonic oscillator group is used to give elegant derivations of identities involving Hermite, Laguerre, Bessel, and hypergeometric functions.

title: Lie theory and separation of variables. 7. The harmonic oscillator in elliptic coordinates and Ince polynomials
abstract: As a continuation of Paper 6 we study the separable basis eigenfunctions and their relationships for the harmonic oscillator Hamiltonian in two space variables with special emphasis on products of Ince polynomials, the eigenfunctions obtained when one separates variables in elliptic coordinates. The overlaps connecting this basis to the polar and Cartesian coordinate bases are obtained by computing in a simpler Bargmann Hilbert space model of the problem. We also show that Ince polynomials are intimately connected with the representation theory of SU (2), the group responsible for the eigenvalue degeneracy of the oscillator Hamiltonian.

title: Complete sets of commuting operators and O (3) scalars in the enveloping algebra of SU (3)
abstract: We consider the ``missing label'' problem for basis vectors of SU(3) representations in a basis corresponding to the group reduction SU(3)⊃O(3)⊃O(2). We prove that only two independent O (3) scalars exist in the enveloping algebra of S U (3), in addition to the obvious ones, namely the angular momentum L2 and the two SU(3) Casimir operators C(2) and C(3). Any one of these two operators (of third and fourth order in the generators) can be added to C(2), C(3), L2, and L3 to form a complete set of commuting operators. The eigenvalues of the third and fourth order scalars X(3) and X(4) are calculated analytically or numerically for many cases of physical interest. The methods developed in this article can be used to resolve a missing label problem for any semisimple group G, when reduced to any semisimple subgroup H.

title: Lie theory and separation of variables. 11. The EPD equation
abstract: We show that the Euler–Poisson–Darboux equation {∂tt−∂rr−[(2m+1)/r]∂r}Θ=0 separates in exactly nine coordinate systems corresponding to nine orbits of symmetric second‐order operators in the enveloping algebra of SL(2,R), the symmetry group of this equation. We employ techniques developed in earlier papers from this series and use the representation theory of SL(2,R) to derive special function identities relating the separated solutions. We also show that the complex EPD equation separates in exactly five coordinate systems corresponding to five orbits of symmetric second‐order operators in the enveloping algebra of SL(2,C).

title: Superintegrability on the two-dimensional hyperboloid. II
abstract: In this work we examine the basis functions for classical and quantum mechanical systems on the two-dimensional hyperboloid that admit separation of variables in at least two coordinate systems. We present all of these cases from a unified point of view. In particular, all of the special functions that arise via variable separation have their essential features expressed in terms of their zeros. The principal new results are the details of the polynomial bases for each of the nonsubgroup bases, not just the subgroup spherical coordinate cases, and the details of the structure of the quadratic symmetry algebras.

title: Separation of variables in Einstein spaces. I. Two ignorable and one null coordinate
abstract: Computes and classifies all Einstein spaces and their corresponding coordinate systems such that the Hamilton-Jacobi equation admits a separation of variables with two ignorable and one essential null coordinates. The authors find that all such spaces correspond to plane-fronted gravitational waves of Petrov type N. They also give necessary conditions which a general space-time must have in order to admit coordinates of the type studied. Furthermore, the integrability conditions for Helmholtz (Schrodinger) separability are completely solved.

title: Improved search for Muon-neutrino to electron-neutrino oscillations in MINOS.
abstract: We report the results of a search for ν(e) appearance in a ν(μ) beam in the MINOS long-baseline neutrino experiment. With an improved analysis and an increased exposure of 8.2 × 10(20) protons on the NuMI target at Fermilab, we find that 2 sin(2) (θ(23))sin(2)(2θ(13))<0.12(0.20) at 90% confidence level for δ = 0 and the normal (inverted) neutrino mass hierarchy, with a best-fit of 2sin(2) (θ(23))sin(2)(2θ(13)) = 0.041(-0.031)(+0.047) (0.079(-0.053) (+0.071)). The θ(13) = 0 hypothesis is disfavored by the MINOS data at the 89% confidence level.

title: Search for Lorentz invariance and CPT violation with the MINOS far detector.
abstract: We searched for a sidereal modulation in the MINOS far detector neutrino rate. Such a signal would be a consequence of Lorentz and CPT violation as described by the standard-model extension framework. It also would be the first detection of a perturbative effect to conventional neutrino mass oscillations. We found no evidence for this sidereal signature, and the upper limits placed on the magnitudes of the Lorentz and CPT violating coefficients describing the theory are an improvement by factors of 20-510 over the current best limits found by using the MINOS near detector.

title: The Cleo III silicon vertex detector
abstract: The CLEO collaboration has embarked on an ambitious upgrade of the CLEO detector to match the upgrade of the CESR accelerator to luminosities of 2 × 10 33 cm −2 sec −1 . This will bring a factor ten improvement in the measurement of rare B decays and hence, increased sensitivity to physics at high mass scales and also the possibility of measuring CP violation. The upgrade includes a new four layer double-sided silicon detector. Each double-sided silicon wafer is 2.6 cmx 5.1 cm × 300 μ m . To maximize solid angle coverage with minimal material in the detector volume the detector chains are very long, up to 5 detectors per electronic readout channel, resulting in large series resistance and large detector capacitance. Moderate backgrounds permit the electronics to operate with a 1–3 μ s shaping time. A new front-end low noise preamplifier is being developed to maximize the signal to noise ratio, along with a modified SVX chip used for analog to digital conversion and sparsification. The detectors are connected to the electronics hybrid by a thin double sided kapton flex circuit. The length of the detector chains also requires novel mechanical support structures — CVD diamond beams axe the baseline solution. We are currently near the end of the prototype stage for all aspects for the project and will move into production by late summer. The detector will be installed in early 1998.

title: Surface underground coincidences at the Soudan mine
abstract: A 37 m{sup 2} surface array which has been operating in coincidence with the Soudan 1 underground experiment is described. A comparison is made between measured and expected underground-surface coincident events. The surface array is used to tag'' the energy of the cosmic ray primary. Distributions of event rates and multiple muon rates as a function of energy are shown. Expected rates and plans for an enhanced array and for coincidences with Soudan 2 are also discussed. 1 ref., 6 figs.

title: Studies of mu's Underground with the Soudan 2 Tracker