Lie theory and separation of variables. 8. Semisubgroup coordinates for Psi tt - Delta 2Psi = 0

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: 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: 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. 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: The SOUDAN 2 detector: The design and construction of the tracking calorimeter modules
abstract: SOUDAN 2 is a 960-ton tracking calorimeter which has been constructed to search for nucleon decay and other phenomena. The full detector consists of 224 calorimeter modules each weighing 4.3 tons. The design and construction of the modules are described. The modules consist of finely segmented iron instrumented with 1 m long drift tubes of 15 mm internal diameter. The tubes enable three spatial coordinates and dE/dx to be recorded for charged particles traversing the tubes.

title: Measurement of the underground atmospheric muon charge ratio using the MINOS Near Detector
abstract: The magnetized MINOS Near Detector, at a depth of 225 mwe, is used to measure the atmospheric muon charge ratio. The ratio of observed positive to negative atmospheric muon rates, using 301 days of data, is measured to be 1.266±0.001(stat)_(-0.014)^(+0.015)(syst). This measurement is consistent with previous results from other shallow underground detectors and is 0.108±0.019(stat+syst) lower than the measurement at the functionally identical MINOS Far Detector at a depth of 2070 mwe. This increase in charge ratio as a function of depth is consistent with an increase in the fraction of muons arising from kaon decay for increasing muon surface energies

title: Testing Lorentz invariance and CPT conservation with NuMI neutrinos in the MINOS near detector.
abstract: A search for a sidereal modulation in the MINOS near detector neutrino data was performed. If present, this signature could be a consequence of Lorentz and CPT violation as predicted by the effective field theory called the standard-model extension. No evidence for a sidereal signal in the data set was found, implying that there is no significant change in neutrino propagation that depends on the direction of the neutrino beam in a sun-centered inertial frame. Upper limits on the magnitudes of the Lorentz and CPT violating terms in the standard-model extension lie between 10(-4) and 10(-2) of the maximum expected, assuming a suppression of these signatures by a factor of 10(-17).

title: Neutrino and antineutrino inclusive charged-current cross section measurements with the MINOS near detector
abstract: This thesis presents the measurement of energy dependence of the neutrino-nucleon inclusive charged current cross section on an isoscalar target in the range 3-50 GeV for neutrinos and 5-50 GeV energy range for antineutrinos. The data set was collected with the MINOS Near Detector using the wide band NuMI beam at Fermilab. The size of the charged current sample is 1.94 x 10{sup 6} neutrino events and 1.60 x 10{sup 5} antineutrino events. The flux has been extracted using a low hadronic energy sub-sample of the charged current events. The energy dependence of the cross section is obtained by dividing the charged current sample with the extracted flux. The neutrino and antineutrino cross section exhibits a linear dependence on energy at high energy but shows deviations from linear behavior at low energy. We also present a measurement of the ratio of antineutrino to neutrino inclusive cross section.

title: RESONANCES IN THE SCATTERING OF ELECTRONS FROM ATOMS.